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CO >- CO [=OU 166429 J M E M O I R S OF THE AMERICAN MATHEMATICAL SOCIETY NUMBFR DECOMPOSITIONS OF OPERATOR ALGKBRAS I and II BY I K. SKOAL PUBLISHED BY THE AMERICAN MATHEMATICAL SOGKTY 531 West J16th St., New York Ciry 1951 OSMANIA UNIVERSITY LIBRARY Call No. 5 (0 & PI ^> ' Accession No . 1 This book should be returned on or before the date last ma DECOMPOSITIONS OF OPERATOR ALGEBRAS. I By I. E. Segal of the University of Chicago 1. Introduction* We show that an algebra of operators on a Hllbert space can be decomposed relative to a Boolean algebra of invariant subspaces as a kind of direct integral, similar to the decomposition as a direct sum of algebras of linear transformations on finite-dimensional spaces. This decomposition results from an interesting decomposition formula, for the "states" of operator algebras which we have treated in [8]. If the Boolean algebra is maximal, and with a certain separability restriction, the constituents In the direct integral are almost everywhere irreducible. It follows that in the case of a separable Hilbert space, a weakly closed self-adjoint algebra is a direct integral of factors. Any continuous unitary representation of a separable locally compact group is a direct integral of irreducible such representations. If G ia uni- modular, then its two-sided regular representation is a direct integral of irreducible two-sided representations. Any measure on a compact metric space which is invariant under a group of homeomorphisms of the space is a direct integral of ergodic measures. Our basic results are closely related to results of von Neumann in The decompositions obtained by von Neumann are from a formal view- Received by the Editors on March 1, 1950. 2 I. E. Segal point nearly Identical with ours, but there are important technical differ- ences in the approached as well as in the results which allow us to give considerably simpler proofs of the key theorems, and which yield a theory better adapted to the study of group representations than that of von Neumann* These differences are notably, first, the use of states, and second, the use of perfect measure spaces, rather than a measure space over the field of Borel subsets of the reals. Bach of these features simplifies the serious measurability problems involved in obtaining decompositions. The concept of direct integral of Hilbert spaces is awkward because the Hilbert spaces may vary in dimensionality, and it is unclear to begin with how a measurable function to such Hilbert spaces should be defined. As a state is a numerical-valued function, there is no such awkwardness about direct integrals of states, and by virtue of the known correspondence between states and representation Hilbert spaces, a decomposition of a state as such an integral induces a decomposition of the Hilbert space into "differential" Hilbert spaces, so to speak. The utilization of perfect measure spaces (on which every bounded measurable function is equivalent to a continuous function) eliminates the need for various kinds of sets of measure zero which occur in von Neumann's theory, and greatly facilitates the reduction of group representations. Our theorem concerning maximal decompositions bears the same formal relation to a theorem of Mautner [5] that our decomposition theory does to that of von Neumann, but the logical roles of these two theorems are very different, as we use our result to de- compose a general algebra of operators into factors, while Mautner 1 s result is derived directly from von Neumann's decomposition theory for general operator algebras. By virtue of the difference between our basic tech- niques (some of which apply to Inseparable spaces) and those of von Neumann, our proofs are for the most part necessarily of a different character from those of von Neumann, and in particular no use is made of the theory of DECOMPOSITIONS OF OPERATOR ALGEBRAS. I 3 analytic sets. 2. Definitions and notations. We introduce here a number of terms and symbols which we shall use without further reference in the re* mainder of the paper. Definition 2.1, A W-algebra (or C*-algebra) is a weakly (or uni- formly) closed self-adjoint (SA) algebra of (bounded linear) operators on a Hilbert space. The term "operator" will always mean "bounded linear oper- ator". For any algebra Ci of operators on a Hilbert space, the set of all operators which commute with every element of d is called the commutor of (L and denoted by ft. 1 . A W*-algebra Ct which contains the identity operator, always designated by I, and is such that Ct ^ Q,' consists (only) of scalar multiples of I is called a factor. The term Hilbert space will be used in the present paper to denote a complex (generalized) Hilbert space of arbitrary dimension (> 0). Definition 2.2 A measure space is the system composed of a set R, a cr-ring ft of subsets of R, and a countably-additive non-negative- valued function r on #*. Such a space is finite if R e H and r(R) is finite. Such a space, denoted as (R, # , r), is called regular if R is a locally compact topological space, 7 is the G'-ring generated by the compact subsets of R, and if for every S /f 9 r(S) * L.U.B. Kc o r (*>) 28 G.L.B.^ _., s r(W) , where K varies over the compact and W over the open sets in l We denote such a space as (R, r), and a countably-additive complex- valued function on ^ is called regular if the positive and nega- tive constituents of its real and imaginary parts are such that the corres- ponding measure spaces are regular (1. e. are regular measures) A finite measure space (R, 1t 9 r) is called perfect if it is regular, and if, furthermore, for every bounded measurable function on the space there is a unique continuous function on R equal almost everywhere to the given 4 I. E. Segal function* Definitions 25 A state of a C*-algebra CL is a linear func- tional cJ on d such that <U (U*U) * and cO(U) = o)(U) for U CL (a bar over a numerical-valued function denotes the complex-con- jugate function), and with L.U.B. ,, rl cJ(U*U) 1, The asspci- II U (I = 1, U e CL ated representation <f> of Q, , gilbert space 7^ , canonical mapping >f of 0. into /V" , and wave fun c t i on z, are the essentially unique objects with the properties: 1) <f> is a representation of Q, on 7V" (i.e. it is a mapping on Q. to the operators on // which preserves algebraic opera- tions, including that of adjunction); 2) if is continuous and linear on d to #-, yj(Ct) is dense in M , and (^(U),^(V = <J(V*U) for any U and V in Q ; 5) f (U) ?/(V) ^(UV) for any U and V in & ; 4) z is an element of //" of unit norm such that o) (U) = (^(U)z, z) and ^(U) = Uz for Ue$. (For further properties and an existence proof, see [8].) A representation ^ of 0. on 14 is called cyclic if there exists an element z in /V- such that <f ( Q. )z is dense in ^ ; such an element z is called a cyclic vector . Definitions 24 The spectrum of a commutative (complex) Banach algebra is the topological space whose set is the collection of all con- tinuous homomorphisms of the algebra into the complex numbers which are not identically zero, and whose topology ia the weak topology in the conjugate space of the algebra. If P is a locally compact Hausdorff space f C(D(or 7? ( P ) ) denotes the Banach algebra of all continuous complex- valued (or real- valued) functions on F 1 which vanish at infinity on T* (a function f on F* vanishes at oo if for every positive number C , the set Cy||f(/)| - & 3 is compact), with the norm of a function taken to be the maximum of its absolute value. Definition 25 If M is a measure space, ^(M) denotes the DECOMPOSITIONS OF OPERATOR ALGEBRAS. I 5 Banach space of otth-power integrable complex- valued functions on M, with the usual norm, where 1 ^ oc * oo , L^lM) designating the Banach algebra of all essentially bounded measurable functions on M. Two functions on a measure space agree nearly everywhere (n. e.) if on every measurable set of finite measure, they agree a. e., and a set in a measure space consists of nearly all points of the space if the intersection of its complement with any measurable set of finite measure has measure zero. If G is a locally compact group, L^ (G) denotes L u (G, m) f where m la Haar measure on G, 3 Decomposition of a state relative o a commutative algebra. We show in this section that any state of a C*-algebra d can be repre- sented as ar. integral of more elementary states, over a measure space built on the spectrum of a given commutative W*-algebra in &' By virtue of the known correspondence between states and representations of C*-algebras this shows, roughly speaking, that every cyclic representation (and hence every representation) of CL is a kind of direct integral of more ele- mentary representations, in such a way that the integrals of the elementary representation spaces are the invariant subspaces of the original repre- sentation space. Thus the present section could be described as an investi- gation of the decomposition of a representation relative to a Boolean alge- bra of invariant subspaces. In a later section the present decomposition, which involves no measurabilit} problems, as states are numerical-valued functions, is used to treat direct integrals of Hilbert spaces (where the dimensionality may vary from point to point) and of operator algebras (which vary similarly), and thereby we avoid the measurabili ty complications inherent in a direct attack on such integrals. It will also be shown later that under suitable hypotheses regarding Ct and the commutative algebra in question, the ele- mentary representations which occur are almost everywhere irreducible. In view of the correspondence between states and representations, 6 I. E. Segal it suffices to consider states o3 of the form o)(T) (Tz, z), where z is a cyclic vector for Q. . We mention also, as is pertinent to the decom- position of an algebra of the form C f , where C is a commutative W*- algebra, that it is known that for any such algebra there exists a family {fy} of mutually disjoint projections in C such that the contraction of (the operators in) C f to Pu.H' has a cyclic vector; and that, moreover, if # is separable, then there always exists a cyclic vector for Q- it- self. THEOREM 1. Let C be a commutative W-algebra on a Hilbert space 7f 9 let $ be a C-algebra in C ' , and let z be_ a normalized cyclic vector for Q. Then there exists a weakly continuous map *'"""* <*V on the spectrum P o C to the conjugate space of L and Perfect measure /m on P such that: 1) for any X e fi. and S C f (SXz, z) y S( /) o)^(X) dfU /), where the mapping S > S() JLs an iso- morphism of C onto the algebra of all complex- valued continuous functions on P j 2) c^^ IJB almost everywhere (relative to (T\^O) a state, and in case Q contains the identity operator, everywhere a state. The proof is based on a series of lemmas, mostly of a measure- theoretic character* LEMMA, 1.1. Let (T\ix) be_ a regular compact measure space, and let ^ be a continuous linear functional on C (P ) Then i V .is the regular countably- additive set function on P cor re a pondi ng to ^ ( i e y(t) *jt( ^) dv(/) for feC(P))^ than for any Borel subset B in P 9 the variation of v over _is We recall the definition of the variation of V , which is a numer- ical function on the Borel subsets of P denoted by Var V : DECOMPOSITIONS OF OPERATOR ALGEBRAS. I (Varv)(B) L.U.B. where is an arbitrary finite ... ^ collection of mutually disjoint Borel subsets of B Now if / f (/)du( /), , is approximated by sums ^ f ( / )u(A ), then as the hm( A ) with feC( absolute value of this sum is bounded, when || f || 5 1, by it follows that (Var/O(B) i L.U.B. [feC(P), ||f||< l] | On the other hand, in a regular locally compact measure space, it is plain that Var yu can be defined by the equation (VaryuHB) L.U.B. V |u( (<,)), where {K} is an arbitrary finite collection of mutually disjoint compact subsets of B . Now let be an arbitrary positive number, let , be mutually disjoint compact subsets of B such that (Varyu)(B) f^ I /*( K j[ )| * C , let ri t (i = 1, , n) be mutually disjoint open subsets of I such that jfl 3 , and with (Varyu)(^i jL - K^ < n" 1 , and let f be an element of C(P) which is 1 on (<., outside of -^-j* and has values between and 1 elsewhere. Setting f(/) ^S^f^ /)sgn/ji( K A )* then || f || f 1 and ) $I l / f A ( / Now and '/B"^- K,)^ by the inequality obtained at the beginning of the proof. It results that > . | LL( |^ )| J | / f( /Jd^M-J /) | * , and hence that (Var/O(B) 5 | / f( /) d/x(/) | +2, which shows that (Varix)(B) ^ ' \XB * - iif ii = i 8 I. Segal LEMMA 1.2. Let P be a compact Hausdorff space, and let yo ! and cr ' be continuous linear f unctionals on C ( F 1 ) such that yO 1 is real and non-negative on non-negative functions, and |c- f (f ) | f <* ^>' ( |f | ) for all f t C ( I" 1 ) and some fixed Qc If /o and cr are the regular countably- additive set functions on P corresponding t /) f and <r f , then cr is absolutely continuous with respect to /O Moreover, if in _ _ j _ accordance with the Radon-Nikodym theorem we set cr ( B ) = / k ( / ) d/}( / ) , " " """""" v 3 where ]3 i H arbitrary Borel subset of P , and k is a_ /p-integrable function, then | k ( / ) | f a almost everywhere with respect to /) Let A be a compact subset of I on which p vanishes and let u A x jO + Var cr Then X is a regular measure on I and hence there exists a sequence \ CL ~\ of open sets in I such that _O. ^ A. L n-' n H^ J0 n ^ 1 , and X(Q ) >A(A). Now let f n be an element of C(D with values between and 1, which is 1 on A and outside of O^ . If 'X. is the characteristic function of A , it results that f ( /) > 'X ( ^ ) aQ with respect to A 9 and hence a.e. with respect to JO and Var cr also. By the Lebesgue convergence theorem, cr ( A ) = llm /f(/)dcr(/) and |cr( A ) | = lini |cr (f ) | < a limsup |/o'(f n ) I = nw/n^ nn nj 11 Cx limsup n |yf n ( /) djO( /) | =ayQ(A) = 0. Thus cy vanishes on any compact set on which yO vanishes. Now let B be an arbitrary Borel subset of P for which ^o(B) = 0. By the regularity of CT there exists a sequence {K^J ^ compact subsets of | such that ^C B and (T( K A ) >cr( B). But CT ( K^) as jO ( K^) - />() = 0, so cr( B) = 0. It remains to show that |k( /)| ^ ex a.e. with respect to jO By Lemma 1.1, ( Var <r ) ( B ) = L.U.B. f c( p } ^ || f || < 1 ly^r( ^ )cl jO( / ) | . Fixing B , let /|^ 1 and jQ ^ be sequences of compact and open sub- sets of P , respectively, such that K n c B c ^\ and X(fl n - K R ) > 0. DECOMPOSITIONS OF OPERATOR ALGEBRAS. I 9 Then if h is a continuous function on F 1 which is 1 on (^ , out- side of ,Q. , and between and 1 elsewhere, h (/) > ^ ( / ) a.e. n p n J3 with respect to A. and we have | / f ( ^)\( / ) d <r ( X ) | > Now |yjf( /)h n ( /) da(af)| - |cr ! (fh n )| < <)i d^o( /). it follows that if fC(P) and ||f|| i 1, then | / f ( / ) dCT(^)| i <y B OtjO(B), and hence (VarcT)(B) f <* jO ( B ) Now it is immediate from a well- known result that (Varcr)(B) = y^ |k( V ) |d jo ( ^ ), and setting P c [/| |k(^)| > Oc 6j, where 6 is a positive number, it results that (Var cr)( P c ) ( Oc f e ) p ( P ). On the other hand, by the preceding argument, (Var(T)(P ) f Oc jO( ? 6 ) , so ( (X + C ) p( ?^ ) < (Var cr)(? c ) < Ocjo(? ). Hence yO ( P e ) =0, and noting that [/( |h(/)| > a] = (J P , , it follows that |k( /)( i oc a.e. with respect to P . n=l n" 1 LEMMA 1 3 Let ( | , /o ) be a compact regular measure space with the following properties: 1) if ff \ is a monotone decreasing sequence of ~ " VP n J - non-negative continuous functions on_ |^ , then there exists greatest low- er bound f of the sequence in T? (P ), and lim / f ( /) d p( /) - r - n %/ n j yf( ^) dp( Y ); 2) :the measure of any nonvoid open set ijs positive^ Then (I , p ) is a perfect measure space Let K be a compact subset of I Then there exists a sequence f n ^ of non-negative-valued continuous functions on \ such that f n ^ X^ ' f ,,( ^> > ^ ^> a * e * wlth ^espect to p , and f ( /) > |r n iv, j n * n vi( ^)* where X v- is the characteristic function of K (such a sequence is readily constructed by induction, using the regular character of p ) Now let f be the G.L.B. of the f in 7? ( V ) As f ( /) > T< ^) n /7 n K a.e., f ( / ) < X^t i) a.e. On the other hand, /f( /) d /0( /) n K /> </ J so that *X k ( ) - fC^)) &f>( ^) 0, from which it follows without difficulty 10 I. E. Segal that T K ( tf) * f( /) a.e. Now let Ti be any Borel subset of I and let IK* T to a mono- tone increasing sequence of compact subsets of B such that p ( (^^) > />(B). Let f 1 e 7?(P) be such that f^tf) = Tt ( /) a.e. Now It is easy to see from the fact that a nonvoid open subset of I has positive measure that the complement of a null set is dense. It follows that as (*!< *)) 2 - ( TL U)) 2 - \ U) = f^ /) a.e., for all *<f 1 <*')) 2 - *.(/), so that f ( /) is everywhere either 1 or 0. Similarly, if I < J, then V 7 " X,, > and ^ results that f 1 (^ / )f.(^) * f,(/ ) *^i M i * J * a.e. so that f ^f . = f.. Hence the sequence "ff*} is monotone increasing, and applying the first condition in the hypothesis of the lemma to the sequence j^- fj} , it follows that the {f^ have a L.U.B. f in ^(P), and that Jt ( * ) djo ( /) > Ji ( / ) djO ( / ) . Since f( /) > t ( 2f), f(/) I Ttv ( /) a.e., which implies that f( if) > ^ ( /) a.e., or f ( ^ ) J X B ( /) a.e. On the other hand, yf( /) dyO( ^) * lim n( /f n ( /) dyo( /) = lim n yo( K n > - />(B) - ,/X^ ( /) d^o( /) , and it follows as in the preceding paragraph that f( ^ ) = lL Q ( tf ) a.e. D Next let f be a bounded measurable function on (f 1 , p). To prove the existence of an element of CtF*) which is equal a.e* to f it is clearly sufficient to consider the case in which f is- real and non- negative. If f is such, say < f ( ^ ) 5 CX for ^eP , then defining M i,n to be the set [V|ot(l - -^ > f ( ^ > - * < l - ^3T> J (i s 0, 1, 2 Jo 2 * * ) M is a measurable set, and if % is the character- i,n i,n istic function of M 1 ^ and if f ^ is in 7t( P ) and equal a.e. to X lf n ' then P uttin f n "Slflo C(1 " ^ )f i,n ' f n e/?( ^>' * and |f( *) - ^ n (^)| f p a.e. and also |f n ^( / ) - f n ( ^) | f a.e. As the complement of a null set is dense, and as f+ - * is in C(P), the DECOMPOSITIONS OP OPERATOR ALGEBRAS. I 11 last inequality holds for all ^ . It follows that S~ , (f ..(zO-f U) ) ^"""n1 n+i n is uniformly convergent, so that f converges uniformly, say to f f , an element of 1? ( P ) It is plain that f ! = f a.e. If there were two elements of C(i ) equal a.e. to f, then we would have a continuous function (their difference) equal a.e. to zero, and by an earlier remark zero everywhere. LEMMA 1.4. With the notation of the theorem, let JJL be the regular measure ojn J 1 such that (Tz, z) - /p T( ^ ) dy^( i ) for T c C, Then ( I , u ) ijJ a_ perfect measure space. Observe first that the measure of a nonvoid open subset of I is positive. For if -O. is a nonvoid open set of measure zero, let K be any nonvoid compact subset, and let f be an element of 7? ( P ) which is 1 on K^ , outside -O. , and between and 1 elsewhere. Then f vanishes outside -fl , so that /(f(^ )) 2 d^x ( / ) 0. It follows that if P Is the operator in C which corresponds to f, then (P z, z) =* || Fz || , and hence Pz = 0. Now if X la an arbitrary element of ui , we have XFz * = FXz. Now Xz | X6 OL ] is dense in ft , and thus P vanishes on a dense set; being continuous, it must vanish identically. Hence f = 0, a contradiction. It remains only to show that the hypothesis of Lemma 1 3 regarding monotone sequences of functions in ~7f(p) is valid. Let {f } be a mono- tone decreasing sequence of non-negative continuous functions on I , and let P be the operator in C which corresponds to f As C is n n algebraically isomorphic to C(P) ( in an adjoint-preserving fashion), its subring ft of self-adjoint elements is order- isomorphic to ^(P) f and we have P n ^ ** n +i > ** n i 0. Obviously tho F R commute with each other and by a known result in the theory of Hilbert space, the strong limit of the sequence ^F n } exists, say lim P x Fx for x H . If 12 I. . Segal f is the element of C(P) corresponding to F, from the inequality P n > F we have f n J f . Now if h e tf(P) and f R > h for all n, then P i H for all n, where H is the element of 7? corresponding to h This would imply that (F n *, *) - (Hx, x) for xeH-, and taking lim^ on both sides of this inequality yields the inequality (Px f x) i (Hx, x) This means that P . H, so that f * h, and hence f is the G.L*B. of the f . Moreover, we have lln^ Jf n ( ^ ) d JUL ( / ) = lim^ (F n z, z) = (Fz, z) LEMMA 15 If o3 1^ a state of a C~algebra containing a normal operator S, then |^(S)| fJ(|S|). We explain that |S| is the absolute-value function, applied to S by the usual operational calculus. The C*-algebra generated by S is isomorphic to the algebra of all continuous complex-valued functions vanishing at Infinity on some locally compact Hausdorff space . *-* s By the Riesz-Markoff theorem, a) (T) = t yv t (5 ) d ^ ( ^ )* for a11 T ln that algebra, t(J^ ) being the corresponding element of C(^ ) and cr being a regular measure on /.rT, Putting s for the element of C, ( ?-* ) corresponding to S, we have o> (s) = A- s(^ ) d cr ( ^ ) and hence ^ PROOF OF THEOREM* We first prove the theorem for the case in which 0. contains the identity operator I. Let u) be the state of Q defined by the equation a) (x) = Uz, z), Xe$. Then ")(SX), as a function of S C for fixed X e $ , is clearly linear, and it is bounded for |o>(sx)| - II SX || < || S|| ||X|| . This linear functional induces, via the correspondence between C and C (P) a bounded linear functional on C(P), and by the known form for such a linear functional we have / c /v, S( ^) d/x. ( o ), for some unique countably-additive regular ' X set function on P , ju We observe next that |<*J (SX) | f <V^)(|S|) for DECOMPOSITIONS OP OPERATOR ALGEBRAS. I 13 X e Ci and S e C , where Oc || x 1 || + || Xg| , X ^ + iXg , 1^ and Xg being SA. For |oJ(SX)| * |^(ax.) + i">(SX Q )| i H (SX ) | + |a>(SX )|; 1 <& 1 6 since the X (J = 1, 2), S, and their ad Joints mutually commute, X.S is normal so by Lemma 1.5 |^(SX )| f ">(|SX |). Now the C*-algebra - generated by X and S is algebraically isomorphic to C ( ^L* ) for J f j some locally compact Hausdorff space /"% 9 and it is easy to deduce from this that |SX. | |S| |X | < || Xj || |S|. It follows that |u)(SXj)| * o3(||X.|| |S|), and the stated inequality follows. Hence Lemma 1.2 applies and asserts that yx is absolutely con- tinuous with respect to /*- /*-r> and tnat If we write (by the Radon-Niko- dym theorem) ^MvC B ) s /^ k x^ ^' d /^( ^ ) * B being an arbitrary Borel subset of I , then k,, is essentially bounded with respect to M . Now by Lemma 1.4, (f 1 ,^) is perfect and hence there exists a unique continu- ous function ki( ^ ) on I which coincides a.e. with respect to M , with k^( fl" ) . Defining c^)^ as the functional on Q. given by the equation u)/(X) = kA(/), then for each Xe&, <*V(X) is continuous function of / . Now as c^ (SX) is linear in X for fixed S, ^ x is a linear function of X, and it follows that k^ y ( /) k^( /) + k^( /) a.e. on (1 >u) for any X and Y In d . Hence k^+y ^ ^^ * k* ( / ) + kA(dO a.e., but as both sides of this equation represent con- tinuous functions of rf , by a familiar argument we have equality every- where. In a similar fashion it follows that for any complex number ft 9 k ! /i ( /) = /3k* ( /) for all / f 1 . These results mean that for each Y 9 <^x is a linear functional on uL . Since the product of two com- muting SA nonegative operators is a SA nonegative operator, ^ (SX) is a positive linear functional, of S, for fixed SA non-negative X, and hence /^ x is then a measure. It follows that i'or such X, k^( ^) is a.e. nonegative, so k^( / ) is everywhere nonegative, and the same is true of oJy (X). To show that <^V is for each / a state of & it remains 14 I. B. Segal only to show that ^(1) =* 1 for all / . Now (Sz, z) *ys(/)d^( /) f&( J ) <*V(I) d^( /), so f$( / ) (l - ^/(I)} dy*( /) a 0, for all S e C . As S ranges over C , S( . ) ranges over all continuous, and so, as (F,^Ji) is perfect, all bounded measurable, complex-valued functions on F, and therefore oiy(l) = 1 ae., but as ^V (I) is a continuous func- tion of if we have equality everywhere. This completes the proof for the case led, for the continuity of ^/ as a function of ^ with values in the conjugate space of CL is equivalent to the continuity of &V(X) as a function of tf for all Xe & Now suppose that I is not necessarily in d , and let Q be the algebra obtained by adjoining I to Q. ; it is not difficult to see that d is a C*-algebra. Then Q C C' and by what has Just been proved there exists a continuous map 7^- > <*)-{ on I to the state space of Q (i.e.jthe collection of all states of d topologized by the weak topology on its conjugate space) such that for X 6 Q and S c C, .(SXz, z) s /i S( /) tJ^(X) dyji( /). It remains only to show that the con- traction ^T-/ of **V to d is a.e. a state, for it is clear that the mapping ^ > <*V is continuous on i to the conjugate space of Q-* Let {v^} be an "approximate identity" for fl(cf. [8]), i.e., ||Vj| 5 1 for all yu, V^efi, and V^. X > X for all X e a. Then for any X and Y in Q , (V^ Xz, V^x Yz) > (Xz, Yz). Thus the equation (V^ x, V^ y) > (x, y) holds for a dense set of x and y in fa 9 and as ||Vj| is bounded, it must consequently hold for all x and y in fr In particular (V^x z, V^z) > (z, z), and so there exists a sequence {u n | in Q. ( a subsequence of the Vu.) such that II ^ II - 1 and (U n z, U n z) > 1. It follows from an equation above that d/A(/) ' > 1, or ^/pCL - u)^(UU)) d^i ( ^) > 0.. As 1 - kV^^n^n) - tne sequence of functions of ^ 9 1 - ^Vf^}^) con ~ verges to zero in L.{ I , Ln) If |U n } is a subsequence such that DECOMPOSITIONS OF OPERATOR ALGEBRAS. I 15 1 - cJ/(U U n ) > a. e. on (/*), then we have L.TJ.B^ cJ^(U U n ) 1 a.e., which shows that t*V is a state a.e. 4. Direct integrals of Hilbert spaces. In this section we define and treat direct integrals of Hilbert spaces, and show that every state decomposition such as that of the preceding section gives rise to this kind of direct integral. In this way an arbitrary C*-algebra fL can be de- composed with respect to any commutative W*-algebra in < (or alter- natively, with respect to any Boolean algebra of closed invariant subspaces). Our definition of direct integral is somewhat similar to that given by von Neumann [16], but we find it necessary to consider two types of integrals, a "strong" and a "weak" type, whose relationship is analogous to that of strong and weak integrals of vector-valued functions. Definition 4.1. Let (R, K. , r) be a measure space M, and suppose that for each point peR there is a Hilbert space ty . A Hilbert space "#" is called a direct integral of the ~/^ over M (sym- bolically 1+ ~J # dr(p) ) if for each x e # there is a function x(p) on R to \J fy , such that x(p):7 c /' , and with the following prop- pe R p p erties (1) and either 2a) or 2b)): 1) if x and y are in #- and if z fcx + /3y, then (x(p), y(p)) is integrable on M, (x, y) J^ U(p), y(p)) dr(p), and z(p) <*x(p) + (3y(p) for almost all p eR; 2) if z(p)CH for all p, if (x(p), z(p)) is measurable for all xe >/; P and if (z(p), z(p)) is integrable on M, then there exists an element z 1 of n* such that a) z v (p) * z(p) almost everywhere on M, or to) (z f P)* x(p)) (z(p), x(p)) almost everywhere on M, x e^. The integral is called strong or weak according as 2a) or 2b) holds. The function x(p) is called the decomposition of x, and we use the following 16 I. . Segal notation for this: x = J~ x(p) dr (p). A linear operator T on 1r is said to be decomposable with re- spect to the preceding direct integral if there is a function T(p) on R to LJ _ fo/ , where ~& is the collection of all bounded linear opera- p e K p p tors on ^*p* such that T(p) 6 T3 for all p and with the property that for all x and y in ^, (T(p) x(p), y(p)) is integrable on M and J (T(p) x(p), y(p)) dr(p) (Tx, y). The function T(p) is then called the decomposition of T, and we symbolize this situation by the notation T s J- T(p) dr(p) If T(p) is almost everywhere a scalar operator, T is called diagonali zable The basic theorem of this section asserts that a state decomposi- tion such as that of the preceding section induces a decomposition of the Hilbert space as a direct integral, in which every element of CL ^ s ^ e - compo sable, and in which the diagonali zable elements are exactly those in C/ Before giving a precise statement of this theorem we make two remarks. First, it la not difficult to show that in case ty is separable, a weak di- rect integral becomes an essentially equivalent strong one when the i^ are replaced by appropriate closed linear subspaces of themselves* Second, the analog, of condition 2b for direct integrals of spaces, in the case of direct integration of operators, is valid without further assumption: if T(p)e 13 for all p, if H T(p)|| is essentially bounded on M, and if P p the integral J (T(p) x(p), y(p)) dr(p) exists for all x and y in ty 9 then there exists an (obviously unique) bounded linear operator T such that (Tx, y) y(T(p) x(p), y(p)) dr(p) for all x and y in ^ For setting <^(x, y) x j(T(p) x(p), y(p)) dr(p), It is clear that Q is conjugate-bilinear (linear in x and con Jugate- linear in y), and that, setting OC * ess aup neR || T(p) || , |Q(x, y) | < /<T(p) x(p), y(p))|> dr(p) i r 2 r 2 i x / 2 x(p)| |y(p)||dr(p) < ^(yilx(p)lr dp(p)) (yi|y(p)|rdr(p))J *ll II 711 9 80 ^ i8 Bounded. It follows readily from the Riesz rep- DECOMPOSITIONS OP OPERATOR ALGEBRAS. I 17 resentation theorem for linear functionals on 1^ that an operator T with the stated property exists, THEOREM 2, Let Q 9 C , and z be a^s in Theorem 1, and let M = (R, A , r) be a measure space with the properties; 1) C i alge- braically isomorphic (in a fashion taking adjoin ts into complex conjugates) with the algebra of all complex* valued bounded measurable on M, the ele- ment S f corresponding tp_ the function S( . ); 2) for each p eR there is a state ^ of Q, , and for T Q, a) (T) is measurable on M; p ^ P 3) for T& and S e C, (STz, z) * / a) (T) S(p) dr(p). c/ R p Then J.f ty , 00 , yj 9 and z are the representation apace, P ' p ( p P representation f Q , canonical map of d into ^/ D f and wave function, > we_ have weakly, and in case Q. JIB respectively, associated with separable in the uniform topology , strongly, ty / //* dr(p) in such a way that for U Q 9 U = f ^(U) dr(p), and Uz J^ >J(U) dr(p) . Every operator decomposable with respect to this direct integral jls jln (^ 9 and an operator i_ diagonalizable i and only If ijt ^a in C We begin b> defining x(p) (more precisely, a residue class of the space of functions on R to \J ty 9 with x(p) 7^- for all p, p P p modulo the linear subspace of functions a.e, zero), for x of the form Uz, by the equation x(p) ^ (\j) TO see that x(p) is single-valued, suppose that Uz = Vz, with U and V in u . Then Wz 0, where W U - V, so (Wz, Wz) 0, but by 3) in the hypothesis, (Wz, Wz) (WWz, z) J a) (WW) dr(p) 0. Now a) (W*W) J for all p, so that the last equation implies o) (W*W) = a.e. This means that ( v (W). in (W)) P ( P IP a.e., or W (W) a.e. on R, and hence x(.) is unique (modulo the subspace mentioned). Now let x be arbitrary in ty and let u n \ be a sequence in Oi such that U n z > x. Then ( U n z - U m z |( > aa m, n > oo , and 18 I.E. Segal ||TJ n z - U m z|| 2 = ((U n - U m )*(U n - U m )z,z) = j p ((U n - U m )*(U n - U m ) ) dr(p) = j"? P (U n) " ^p( u mH ^(p) - * 0. Y/e now apply the procedure utilized in the proof of the ^lesz-Fischer theorem to the selection of a subsequence of ifp^nM wnose Iknit exists a. e. and defines the function which we shall designate as x(p). Let "(n^V be a subsequence of the positive inte- gers such that n 1+1 > n 1 and with jll >f p (U n ) - ^ p ( u m )| 2 ^r(p) < 8"" 1 for n and m greater than n^ The set of ps for which II Yp( U n ) " ^p^ u n HI > C is, for > O f clearly of measure less than C ~ 2 8"^ t and taking = 2" 1 , it results that ||>/p(U n ) - 7f p (U n+ ) || < 2" 1 except on a set of measure less than 2" 1 . Therefore the inequalities II ^ p (U n ) - >f p ( u n HI < 2" hold simultaneously for all i > j except on a set D. of measure less than Zl^.2" 1 = 2~3" 1 " 1 . It follows that the series 2^1=1 Otp^ni) " "fp^ni+i)} ls ^i^o^ly convergent for p ^ Dj t and hence that lim^ ^(^n ) exists uniformly for p ^ D j . Putting x(p) for that limit, it is clear from the fact that **(Dj) >0 as j *co that the Unit exists a, e. so that x(p) is defined a. e. (and nay be defined arbi- trarily on the null set on w^ich tl e limit fails to exist). Prom the equation ^ || ^ ( u m ) - f p (u n Hpclr(p) < 8" 1 for m and > n , it results that ypll ^ p (U m ) - Yp^nJlPdrCp) < 8" 1 for m and n* greater than n^ f and for any k. Now p^n^ c o nv e^Kes uniformly to x(p) as J ->co, on R-^ k , so </R-D, II Yp^ u n' ~ X (P) If 2 ^(P) < 8" 1 for m > n lf and for any k. Letting k > oo , it follows that </ R ll^ p (U m ) - x(p)||dr(p) < 8"* 1 if m > n so J \\ fp(\) - x(p) || dr(p) as m -co. We show next that the function x(p) is independent of the se- quence (U n ) utilized. Suppose that {u 1 } is a sequence in Q. such that U^a > x f and let x'(p) be a function obtained from {p n l in the same fashion as that in which x(p) was obtained from f^ n l Then dr(p) > as m > CD Thus both DECOMPOSITIONS OP OPERATOR ALGEBRAS. I 19 p<V " * (p) H 2 } and {"YP^ " xl(p) !| 2 } ^nverge to zero L^M). Now J || ^ (U m )- >f p (uMl| 2 dr(p) * | U m z - U m z || 2 > as m > oo , so that illYn^ ^ " 7 n^ f ^ " J also Conver 8 es to zero to I^(M) Choosing a common subsequence {m*/ such that the corresponding subsequences of all three sequences converge a.e., we have || x(p) - x'(p)| . || x(p) - x f (p) || a.e., i.e., x(p) x 1 (p) a.e. If y is any element in "& and if V z > y with V n e 0. , let /V j- be a subsequence such that 7? (V m ) > y(p) a.e. Then a.e. we have (x(p), y(p)) s lim. * (^ (^ ^'^^m^' and 1,3 P i IP j ( >f (U ), >/ (V m )) a) (V* U n ), which is a measurable function of p. vp n i * P 3 P m l 1 Thus (x(p), y(p)) is a.e. the limit of a sequence of measurable functions, and is hence itself measurable. Moreover, (x(p), y(p)) is Integrable and has (x, y) for its integral. To show the integrability, it suffices to show that (x(p), y(p)) is the limit in L^M) of the integrable functions ( >/ (U n ), ^(V )), as i, 3 > oo Now (x(p) f y(p)) - ( P n l (p m 3 p f, y(p)) - ( ^ p (U n ) t (U ), y(p)) - ( >/ (U ), vi (V m ))}, so that the left side of this n^ vp n^ ^p "ij ^ equation is bounded by || x(p) - ^ (U n ) || || y(p) || ^ ll>f n (^ n ) II II y(p) - ( p i p i 1? ( u n 'II Hence, applying Schwarz 1 inequality, J n )| dr(p) < J 1/2 20 I. E. Segal Now y(p) IP "J ~ Yp^ U m * * *f * U m *' S0 llytp)||f II y(p) - >? (U m ) || + J J J 11^ (U n )|| . By Minkowski's inequality, J\\ y(p)|| 2 dr(p) < f /? ,, 2 ") 1/2 f /* 2 1 1/2 / /[I y(p) ~ >7 n ^n ^ II ^ r vP) f "*" 1 /If ^ (U ) U dr(p) / , which shows (J IP"- J li/'P n 1 J / ^^ j that ||y(p)|| 2 is integrable. Also, l\\ v (U )|| 2 dr(p) * / ' *^ i /> 2 /o) (U* J n ) dr(p) = (U* U n z, z) = || U n z|| , which is bounded as i > oo. It is easy to conclude that /|(x(p), y(p)) - ( ^ p ( u n ) ^ ( v m )( * J dr(p) > as i, J > oo This shows that (x(p), y(p)) is integrable and that its integral is lln^ J^^n >* fp* V }) dr(p) . ) (V* U ) dr(p) = Iim 4 (U n z, V m z) = (x, y). That x(.) is J J P mj n i i,3 "i m j a linear function of x is clear from the fact that Oc yi (U ) + fiy ^ V m ' on tlie one hand converges a.e. as i > oo to ^ x(p) +/3y(p), and on the other, equals ^ ( flt U >/5V ), of which a subsequence con- ( P n i ; m j verges a.e. to (0tx - Ay)(p) (for ( OL U - /3V )z > Oc x Ay as ' n i / m j f i, j > CD ). Before concluding the proof that ty is the direct integral of the // , we consider the decomposition of operators. Let T, U, and V be P n arbitrary in Q . Then (TUz, Vz) = (V*TUz, z) = /J (V*TU) dr(p) = P / ,/(>; (TU), }? (V)) dr(p) /( 00 (T) >; (U), w (V)) dr(p). This shows ^ ( P (P c/fp (p (P that the equation (Tx, y) = /( <f> (T) x(p), y(p)) dr(p) holds for x and y of the forms x = Uz, y = Vz. Now let x be arbitrary in /& and let n > x, where /* n } i3 a sequence in LIT. Then if y = Vz, we have and (T) *(p) - x(p), y(p)) dr(p)|, =D say. Now K9MT)U n (p) - x(p)) f y(p))| < II?MT)|| ||x n (p) -x(p)|| fly(p)|| , and DECOMPOSITIONS OF OPShATOK ALGLBhAS. I 21 (T) x(p), y(p)) dr(p)| T) *() - x() )) dr) T) \\<f (T)|| f || T|| , as this is true for any representation, so D f P C r o ) 1/2 ||x n (p) - x(p)|| ||y(p)|| dr(p) < || T || f J\\ x (p) - x(p) |P drtpij 2 "> 1/2 y(p) II dr(p) j , which has the limit zero as n > oo On the other hand, it is plain that ( Tx n * y) > ( Tx * y)* so in this cane we like- wise have (Tx, y) = J( <f (T) x(p), y(p)) dr(p). Next let y be arbi- trary in #" , let fy n } toe a sequence in dz with y n > y. Then esti- mating | J( (f (T) x(p), y n (p)) dr(p) - J( <f (T) x(p), y(p)) dr(p)| as in the case of a similar expression above, it results that the present expres- sion has the limit zero as n > CD It follows that the preceding formula for (Tx, y) is valid for arbitrary x and y in /T" Now suppose that T e CL and S &. We shall show that (STx, y) == ys(p) ( <f (T) x(p). y(p)) dr(p) and that (Sx, y) * t/S(p) (x(p), y(p)) dr(p), for all x and y in J+ For x Uz and y = Vz, the equation (STx, y) (p) (^ (T) x(p), y(p)) dr(p) follows trivially from the hypothesis. Now if x and y are arbitrary in f<^ 9 and if -[x | and {y n } are sequences in dz which converge respectively to x and to y, then (STx, y) lirn^ (STx n , y n ) * lim nv /S(p) ( f (T) ^ n (p), y n (p dr(p). Now S(p) is bounded as a func- tion of p, and this observation together with an argument used above in a similar situation shows that J$(p) ( ^ p ( T ) x n (P> y n (P^ dr(p) > ^(p) (<f> (T) x(p), y(p)) dr(p), aa n > oo . Again, if x Uz with U d and y e # f and putting fw^} for an approximate identity for CL 9 we have (SW^ x, y) * ^(P) ( ^ p (*> } x(p) ' y(p)) dr(p) * (S ^ Uz ' y) ^/s(p) (^ p (v u) z(p) * y (p)) dr(p) * Slnce ^ P ( v u) - > ?y u) uniformly, relative to /x, i.e., llfp^ U) - ^ p (U)ll > uniformly on R, so that a sequence //^i) exists such that f (W^ U) > ^ p (U) 22 I. E. Segal uniformly relative to 1, the last expression converges to u/SCp) (? p (U) z(p), y(p)) dr(p) = j&(v) (x(p), y(p)) dr(p). Now, If x and y are both arbitrary In /V* f let 0^} t> e a sequence In Qz which converges to x. Then (Sx, y) = ^-^"n ^ x n* ^ s ^^ m (x (p) f y(p)) dr(p), which last expression Is readily seen to equal (x(p), y(p)) dr(p). W e observe finally that (Sx)(p) * S(p) x(p) a.e., for ^((Sx) (p) - S(p) x(p),(Sx)(p) - 8(p) x(p)) dr(p) ), (Sx)(p)) - (S(p) x(p), (Sx)(p)) - ((Sx)(p), S(p) x(p)) -t- (S(p) x(p), S(p) x(p))] dr(p) (Sx, Sx) - (Sx, Sx) -fs(Sx) f x) + ((SS)x, x) (the assumption that the integral exists being Justified by the given expansion of the integrand). We now conclude the proof that ft* is a direct integral of the # Suppose that w ! (p) is a function on R such that w'(p) e ^/" for pR, (W f (p) f w'(p)) Is integrable on M, and with (x(p), w'(p)) measur- able on M for all x #. Then (x(p), w'(p)) is integrable on M f for by two applications of Schwarz 1 inequality we have ), w'(p))|dr(p) f JxlpJH || w'(p)|| dr(p) < The same inequality shows that setting L(x) * /(x(p), w'(p)) dr(p) f then L is a continuous linear functional on rr Hence there exists an element w e ~H- such that L(x) (x, w) . It is obvious that y(x(p), w(p) - w(p)) dr(p) = for all x e M Putting x Sy with S and recalling that (Sy)(p) - S(p) y(p) a.e., there results the equation yStp) (y(p)* w'(p) - w(p)) 0. As S ranges over C 9 &() ranges over the space of all bounded measurable functions on M, and it follows that (y(p), w'(p) - w(p)) * a.e., or (w'(p), y(p)) (^(p) y(p)) a.e., i.e v condition 2b) in the definition of a direct Integral of Hilbert spaces is valid. If Q is separable, say with \U.; 1=1,2, .. .,j dense, then the >7 (U.) are dense in #- f and as DECOMPOSITIONS OP OPERATOR ALGEBRAS. I 23 (w'(p)f yj (t^)) s (w(p), -y (1 V) simultaneously for all i, a.e., it results that w*(p) w(p) a.e. so that the integral is strong, It remains to show that if an operator T is decomposable or dla- gonalizable, then it is, respectively, in C* or C. Now suppose that T is decomposable, so that (Tx, y) = y(T(p) x(p), y(p)) dr(p) for all x and y in Jr > and some function T(p) such that T(p) is an oper- ator on H for p e R, and with || T(p) || essentially bounded on M. It S is arbitrary in C , we have (TSx, y) (p) (Sx) (p) , y(p)) dr(p) - y(T(p)S(p)x( P ), y(p)) dr(p) i /T(p)x(p) f S(p)y(p)) dr(p) - y?T(p)x(p), (Sy)(p)) dr(p) - (Tx, Sy) - (STx, y). Hence ST TS or T e C f . It is trivial to show from the fact that C is isomorphic with the algebra of all bounded measurable functions on M, that a diagonalizable operator must be in C % The proof of Theorem 2 is thereby concluded, and we have incidental- ly established the following corollaries. COROLLARY 2.1. If S e C , T e<2 , x e #-, and it x(p) is the de- composition of x, then the decomposition of Sx _is S(p)x(p) and the de- composition of Tx is 00 (T)x(p) - _ r p COROLLARY 2.2, If x j[ e #-( i = 1, 2, . . . ) and x t > x, then there exists a subsequence fn<) of the positive integers such that x (p) > " v ** - - - - - * n x(p) a.e, We close this section by obtaining a result which will be useful in the treatment of separable Hllbert spaces. THEOREM 3. With the notation of Theorem 2, let, Q be separable (IE the uniform topology) . Then every strong limit of a sequence of oper- ators in (2 is decomposable relative ^ the decomposition of /^ des- cribed in Theorem 2. 24 I. E. Segal Let {T 1 be a sequence of operators on Q which converges n J strongly to an operator T; It must be shown that T is decomposable. Let {Ujl be a countable dense subset of Q , and set x = U z; then (x,} is dense in 1^ By Corollary 2.2, there exists a subsequence in, ,~V of the integers such that T n (p) x (p) converges a.e. to (TxXpK Next, there exists a subsequence {n^ 2 } or the n such that T n (p)x (p) i,2 converges a.e. to (TXg)(p). Proceeding in this fashion by induction, and employing the Cantor diagonal process, it follows that there exists a sub- sequence n | of the integers such that T n (p)x (p) converges a.e. as i J i > oo to (Tx J )(p) Now as the U are dense in <2_ , the 77 (U.) J 3 (p J are, for each p e R, dense in ^ Moreover, || T || is bounded because {T | is strongly convergent, arid hence || T (p)|| is bounded for peK and i = 1, 2, ... A bounded sequence of operatorc which converge on a dense set is strongly convergent, and hence ( T- (p)r has a.e. a strong l n J limit T(p). It is clear that T(p) x.(p) = (Tx )(p) a.e. It follows easily r that (Tx, y) = /(T(p) x(p), y(p)) dr(p) in the special case in which is one of the x and y is arbitrary in 7^. Now if x is arbitrary in j 1*r and if fx 1 } is a subsequence of the x. such that x f > x, then J J J clearly (Tx ? , y) > (Tx, y) and on the other hand * f (p) - x(p)|| 2 dr(p) > 0. Now ||T(p)|| < lirasup || T (p) || < ||T||, J i n i so that (noting that (T(p) x(p), y(p)) is measurable on M, being a.e. equal to lim (T (p) x(p), y(p)) ) i n x (p), y(p)) dr(p) - T(p) x(p), y(p)) dr(p) | < * J ,/f(T(p) x'(p) - x(p), y(p))| dr(p) < ^|T| ||x'(p) - x(p)|| || y(p) || dr(p) J J DECOMPOSITIONS OF OPERATOR ALGEBRAS. I 25 - II T|| || x' - x|| || y|| > 0. It follows that the preceding equation for (Tx, y) holds for arbitrary x and y in ^. Definition 4.2. If T is the strong limit of a sequence { T } in Q such that {T n (p) 5 f (T )} converges strongly for almost all peR to T(p), then T(.) is called the canonical decomposition of T (with respect to Q , & , and z), Remark 41, To Justify the preceding definition it should be shown that the canonical decomposition of T is unique. Suppose now that {T f J is a sequence in Q which converges strongly to T and that a.e. on M, T ! (p) converges strongly to T(p) Then a.e., for all U, in the n i dense subset of Q which occurs in the proof of the preceding theorem, *f (VII > 0, and ||T'(p) yj (U ) - 'P 1 n ( P 1 T f (p) -+{ (1^)11 > 0. Now || T n - T^l^zfl 2 * (T n (p) - T(p)) Y (U ) || dr(p), and so there exists a subsequence {n.} J\\ of the integers such that || T (p) 77 (U.) - T* ?/ (n ) II > "< 'P 1 Hj f p 1 J > oo, a.e. simultaneously in i. By Minkowski ! s inequality, > as p n^) - T n (p) ^ptU^II + T n ( P } ^p^i' ' T A ( P> ^p^l)" * I' T A (P) V^' " J J J T'(p) >( (U )|| and it results that || T(p) ^(^^ - T(p) ~^ p (^^ II " a.e* simultaneously in i. As for each p e R, the vi (U.) are dense in >f , it follows that a.e. T(p) = T f (p). COROLLARY 3.1 ,If T and U are strong limits f sequences Jin (2 9 if x e ?/", and ^f OC i a complex number , then the following equa- tions hold a..: (T + U)(p) = T(p) -^ U(p); (TU)(p) = T(p)U(p); (OcT)(p) = ^T(p); (Tx)(p) T(p)x(p). Here T(.) and U(.) are the canonical decompositions of T and U and x() and (Tx)U) are the 26 I. B. Segal decomposition a of x and Tx, respectively. Let {T | and {U n } be sequences in Q, which converge to T and U respectively, and such that {r n (p)j and {^ n (P)} &$ converge to T(p) and U(p) respectively. Then {? n + U n } converges strongly to T + U, and a.e. {T (p) + U n (p)} converges strongly to T(p) + U(p), which shows that (T +U)(p) = T(p) + U(p) (a.e.). Similarly, it follows that (TU)(p) T(p)U(p) and (OcT)(p) = <*T(p). If x is arbitrary in ty, ||(Tx)(p) - T(p)x(p)|| is a.e. equal to lim n || (Tx)(p) - T n (p)x(p)|| . On the other hand, J\\ (Tx)(p) - T n (p)x(p)|| 2 dr(p) = || Tx - T Q X || 2 > 0, and so by Corollary 2.2 there exists a subsequence {n^} of the integers such that || (Tx)(p) - T n (p)x(p)|| > a.e. It follows that || (Tx)(p) - T n (p)x(p)|| =0 a.e., I.e., (Tx)(p) =T(p)x(p) a.e. Remark 4.5. A slight modification of the proof of Theorem 3 shows that every strong limit of a sequence of decomposable operators is itself decomposable. For as Q is separable, Qz is separable, so that fa is separable. If \x,; 1 = 1, 2, . . . ^ is a countable dense subset of fy , then -[x (/); 1 = 1, 2, ...} is a.e. dense in ^^ 9 for by Corollary 2.2, if "T x :i4j ^ 3 a su bsequence of x } which converges to U-z, there is a subsequence of this subsequence whose decomposition function converges a.e. to ^7/(U ), and the >7/(U ) are dense in T 1 /^ . The remainder of the proof is the same. We note finally that as a weak limit of a sequence of operators is a strong limit of a sequence of finite linear combinations of the operators (cf. [l8j), the set of all decomposable operators Is closed in the weak sequential topology, when Q Is separable. 5. Maximal de compos 1 1 ions . The complete reduction of an algebra of linear transformations on a finite-dimensional linear space is determined by the selection of a maximal Boolean algebra of invariant subspaces under DECOMPOSITIONS OP OPERATOR ALGEBRAS. I 27 the algebra. Such a selection is likewise possible In the case of Hilbert space. In fact, It Is not difficult to show that If Q_ is a C*-algebra on a Hilbert space "ht* 9 and If C Is any maximal abellan self-adjoint subalgebra of Q* , then the ranges of the projections in C constitute a maximal Boolean algebra of closed invariant subspaces under A, (Actual- ly, the Zorn principle shows that such a selection is possible on any linear space, but in the case of Hilbert space, it can be made in the foregoing way, with complementation in the Boolean algebra coinciding with orthogonal complementation.) The main purpose of this section is to show that if d is separable, then the components in the reduction of & relative to such an algebra C as in the preceding sections, are a.e. irreducible. That is, the ft are a.e. irreducible under the <f ( Q. ) . A similar result, based on the von Neumann reduction theory, is due to Mautner [6] In the next section we apply our result here to obtain a decomposition into factors of an arbitrary W*-algebra, similar to that obtained by von Neumann, for "rings 11 with respect to their centers. THEOREM 4. With the notations of Theorems 1 and 2, let the state decomposition hypothesized i.n Theorem 2 be that obtained in Theorem 1, S M - ( f 1 , JJL ) and let C & maximal abelian in & ! Then (jP . j^ almost everywhere Irreducible. We first prove a lemma on inverses of continuous maps of com- pact spaces which plays a role somewhat similar to that of a lemma of von Neumann fl6, Lemma 5j concerning inverses of continuous functions on analy- tic sets* IMMA 4.1. Let f be a continuous function from a compact metric space C to a compact metric space D. J-f E 3 - 8 an open subset of C, there exists a Borel function g on f (E) _to C, such that g(y)e f l( y ) n E for yef(E). 28 I* Segal We shall present the proof in stages, first considering the case E = C, then the case in which E is closed rather than open, and finally the general case. This arrangement of the proof is not logically necessary, but seems to clarify its structure. SUBLEMMA 4.1.1, The Lemma la, valid in case E = C. Let {* } he a countable dense set in C such that x^ x^ if n ^ m (n = 1, 2, ...) (we shall assume that C is not finite, as the re- sult is obvious for the finite case). Let S (x) denote the set of all points x 1 e C such that d(x, x f ) < e, where d(x, x 1 ) denotes the dis- tance from x to x f , and e > 0. We define a function g^(y) on D as follows: g (y) = that x which has the least index n among the m such that S..(x ) # f -1 (y), where A # B means that the sets A and B inter- sect. It is clear that such points x^ exist, for otherwise there would exist points in C whose distance from the x m was ^ 1. Now g is a Borel function* To show this it suffices, in view of the circumstance that g is (at most) countably-valued with values among the x , to show that for any n, g (x n ) is a Borel set. Clearly g^^^O is th y such that a) S (x )#f -1 (y), and b) n is the least m such that S-^x )#f -1 (y). Since the assertion S-(x )#f" (y) is equivalent to the assertion yef(S, (x )), it follows that g'^x ) * f (S, (x )) - i m 1 n i n I I f(S_(x )). Now S (x) is a countable union of closed suosets, and ^<n lp G hence a countable union of compact subsets. By the compactness of a con- tinuous image of a compact set, f(S e (x)) is a countable union of compact sets, and so is a Borel set. It results that g" (x ) is Borel. 1 n Next we define g p (y) as that x^ which has the least index m among m for which s -,/ p ( x m W (y) A S (g (y)). By the same argument as before, g 2 (y) exists, and its values are among the x . Now DECOMPOSITIONS OF OPEKATOK ALGEBRAS. I 29 U m [y| g 2 (y) - x n , g^y) = xj |J [y|n is least p such that S, /0 (x ) t~ l (j) ^ S, (x ) / 0] n m l/<d p J. m * Since g is Borel, so is [ylg-y) = x] . Moreover, m [y|n is least p such that S (x ) n f" 1 (y) n Si^) / ] m 1/2 " fyln is least p such that yf(S, / (x ) S, (x ))1 */ ^ P 1 ni f(S , (x ) ^S (x )) - I I f(S / (x ) n S (x )), which is easily seen 1/2 n in ^]3<n ' P 1 ro to be Borel. We note also that d(g (y), g (y)) < 3/2, for if g (y) - x , \ ** ^ n then S 1/2 (x n )#S 1 (g i (y)). Uow v/e define g by induction as follov/s : p; (y) is tlmt ^ r x n \vhich h.-is the Ic.'ist index p such th;.t S ^ (xp) # (f^ty) n S 2 ( <5 x (y))). 1/2 ^ , /^i <: r i fhcn clearly G^ 1 ^^) = [yl G r (y) = x n ] = ^ m t7l n la S 1/2 r-l ( V* (f ~ (y)nS - ^^^ISr-l^' = X [y|g (y) = x ] is Borel by the induction hypothesis, and tne other set involved in the intersection behind (J is [y|n is least p such that II f(S _ (x ) n S (x )), which is Borel. Thus g^U ) is ^p<n 1/2 1 *" 1 P l/2 r " 2 m r n Borel, and hence so is g r . Finally, d(g r (y), ^i^^ <(l/2 r " 1 ) + (l/2 r ~ 2 ), since a sphere of radius l/2 r-1 around g r (y) meets a sphere of raaius l/2 r ~ around g r _-i(y) Clearly the series 5"", ^(g ^ (y) g (y ls convergent, and * r=i r-*"i r hence the sequence {g (y)f converges uniformly to a Borel function g. Now g^y) is of distance less than 1/2 11 " 1 from the closed set f" (y and hence g(y) f"^"(y), concluding the proof of the aublemma. 30 1. E. Segal SUBLEMMA 4.1.2. With the notation f Lemma 4.1, let G be a closed subset f C, and let g be a Borel function on F f (G) t G such that g (y) f (y), for y eF. Then there exists a Borel function g on D Jbo C which coincides ori F with g , and such that g(y) e f (y) for ye D. Let {x } be dense in C - G (which we may assume to be infinite), We define a function g on D as follows: g, (y) = g (y) if yeF, and 1 i o if y ^ F, then g- (y) is that x n such that n is the least p for which S (x )#f~ (y). Then g^^ is Borel, for if K is any closed set in C, g;(K) = g'K G)vJ gCK - G). Now g"(K^G) =g - (K^G), which is 111 1 O a Borel subset of F and hence a Borel subset of D. To show that g'-^K - G) is Borel, it suffices to show that g" 1 (x n ) is Borel, and this can be done Just as in the proof of the 1'irst sublemma. The proof may now be completed by constructing a sequence i^L.} by induction, again just as in the proof of Sublemma ! PROOF OF LEMMA. Let E = ^ n G n* where G +- 3 G and the G n are compact, and let F = f(G ). By Sublemma 2 and induction there exists a sequence h } of Borel functions h on F to G such that h (y) f -1 (y) and h^^ agrees with h n on F n Let z be an arbitrary fixed point in C, and define R on f(E) by setting g^y) = ^(y) for yeF , and g (y) z for y k F . Then it is easily seen that g^ is n n * n n Borel. Now for any fixed yef(E), y F n for sufficiently large n, so that gy) converges, say to g(y), with g(y)ef" (y). It is plain that g is a function satisfying the conclusion of the lemma. The following lemma shows roughly that the ^ / are irreducible "on the average". LEMMA 4.2. With the notation of the present theorem, suppose that P, +/3(V)cy. where O ( /) and fi( /) are measurable DECOMPOSITIONS OF OPERATOR ALGEBRAS. I 31 functions on Y* to the interval (0, 1) such that OL( /) + (3(1^) 1 for all 9< , and where for each Ue Q. , /?/(U) and cy(U) are measurable functions of if 9 p j and ay being in the conjugate space of Q, and states o_f CL Then a.eu If a ( /) - 1/2 we put pj - 2( OC ( /)y^ + ( (3 ( /) - 1/2) and CT^ = G x ^; if ct( /) > 1/2 we put ^ yO^/ and CT^ 2( OL( 1/2)^ + /3(/)cy). Then u) / (l/2)( yoj,+ cr| ), />^(U) and are measurable functions of if for UC Q. , and jo ^ and cr^x are states a.e. Hence it suffices to consider the case in which cx( /) * ft( /) * 1/2. As proved in [8, page 80] (though not stated formally as a theorem) the equation 2o)/ jO^ 4- c^ implies that jO/U) = (T^ ^(U)z^, z/) for Ue a , where TV ( <p f ( A)) 1 and || T^ || 2 2. Now if U and V are in a , (T/ ^ (U), ^/V)) (T r ^ (U)z^ , ^Wz/ ) - (T^ <?/ (V*U)z^ , z^ ) /O|<V*U), and so is a measurable function of / It follows readily that for arbitrary x and y in 14 , (T^ x( / ) ,y ( / ) ) is a measurable function of o Hence there exists an operator T on ?/" such that (Tx, y) = J(T/x(1'), y( / ) d^ ( /) for all x and y in >/ . Aa T is decomposable, T Ci On the other hand, T CL 9 for if U, V, and W are in CL , (TUVz, Wz) (TVz, U*Wz) = (UTVz f Wz). As V and W range over CL , Vz and Wz range over dense subsets of # , and from this it follows that (TUx, y) (UTx, y) for all x and y in 1+ , so that TU UT. Thus T efl'^C^ but by assumption, fl' n C/ C. Now let T(/) be the continuous function on I corresponding to T and let S be arbitrary in C . Then (STx, y) (Tx, Sy) - y(T^x(/), S(/)y(/)) d*x( /) (by Corollary 2.1) ys( / ) (T^ x( / ) , y( / ) ) dyA ( /) . On the 32 I. E. Segal other hand, (STx, y) = ys( ^)T( /)(*( /), y( ^)) d^i( /) . From the arbi- trary character of S it results that (T^x(^), y(/)) =T(zO(x(/), y(/)) a.e. In particular (If ^(U), fy(V)) = T(dO (>^(U), ^(V)) a,e. if U and V are in $. , and if {u3 is any sequence of elements of Q, (Tj, >f^(U 1 ), ^(UJ) = T( /) ( ^(l^), ^(Uj)) holds simultan- eously for all i and J a.e. i^ow assuming the Uj: to constitute a dense subset of 0. , the ?//(Uj_) ar dense in #y , and hence a.e. (T/ x/ , y^/ ) = T( /) (x^ , y^ ) for all x ^ and y^ in //-^ . There- fore T^ - T( /) a.e. so that a.e. jO^ is proportional to <A^ , and as jdf and a) ^ are both states, <*>/ = pj a.e* PROOF Of TliEOI^EM. Let N be a Borel set of measure zero which contains the set of /' s for which cJ^ is not a state. Let -O- be the state space of the C-a-algebra Q_ generated by d and the identity I. Then $ consists of all operators of the form ai + U, with U e d , and so is separable. It follows that -O- is compact metric (if {Uj} is a dense sequence in Q^, d(y},cr) =5"^ 2" 1 || ^ || -1 | yot^) - crC^)) is a metric inducing the weak topology). For any state cO of d , let oJ 1 denoter the extension to Q defined by the equation o) f ( oCi + U) = ot -i- o>(U); then a)' is a state of Q and is pure if and only if cO is pure. That a) 1 is a state is clear with the exception of the requirement that cj'( a I + U) ^ if (XI + U i 0. Now (XI + U > means CCI + U = (/3I + V) 2 with V (2 and /3 I -^ V self-adjoint. We can suppose that ft is real and V = V*, for otherwise the equation /3I + V = ( /3I * V)* implies that I = ( ft - /3 ) (V - V*) so that I e and Q. I = d. Plainly a3\ ^3 I + V) = ft 2 + 2/3a)(V) + c)(V 2 ), which is non- negative for real /3 by the fact that the geometric mean is dominated by the arithmetic mean. It is immediate that u) is pure if 60 f is pure, and the converse is proved in [8] , p. 87, DECOMPOSITIONS OP OPERATOR ALGEBRAS. I 33 Now let C be the product space -Q- X -**-, D the space -^- f the mapping (p 1 , cr 1 ) > (l/2)( /O 1 + cr 1 ), and E the subset of C con- isting of all elements of the form ( jo\ cr* 1 ) with J? cr f . Now C - E s the diagonal set of all (/>>/>') wl th yO'e jfl ; as I~l is compact and he mapping yC>' > ( d, jj) continuous, C - E is compact and hence E s open. It follows that Lemma 4.1 applies and states that there exists a orel mapping ^ on the set JLJL of non-extreme points of -CL to n X n such that if yr(u$) = ( p\ </), then a)' = (1/2) ( p + cr 1 ) . For n extreme point to 1 of li. we extend ^lr by defining ^~(cJ) = u/; X" is then defined on JTL , and is Borel. T show that )^ is Borel ctually, let K be any closed subset of -*i- * -^ L and K-^ the set of 11 elements of K which are diagonal. Then ^>" 1 (K) = ")^" 1 (K )^ 'V / " 1 K - K ). Now K = K n (C - E) and hence is compact; as yQ f > (/0/d) s a ho me onio r phi sin on _Ll> to C - E, and as ~Y on C - E takes /d\jt) into yO f , y" (K ) is compact and hence Borel in -Q, . Now - K is the intersection of E with a closed set and so is closed elatlve to E; as }^ before extension was Borel, 7^" (K - K ) must e Borel. Finally Y (K) is Borel, being the union of two Borel sets. We put ~y/(u) = ((t*) f ), %(C4))), and then and % are ise Borel, being compositions of Borel with continuous functions. Let J e the mapping ^ > a) of states of Q into the continuous linear unctlonals on CL ; then it is clear from the definition of the weak topol- gy that J is continuous. Now let /O^ = j ( u)^ ) anc * Oy** %( a)/- ) for / 4 |\j , so that yO^ and CT^ are states of CL when ^ is a state, and (l/2)(yO,>+ or^ ) = a)/. For any Ue$ f y^/ u ^ and r^(U) are measurable functions on I - N, for taking the case of X)^(U), it is the product of the successive maps (a) o > t*>^ , on I - N to ^ ne state space of CL 5 (b) <) > oJ f on the state space of CL to H ; (c) 0)' > (U 1 ) on JT to /I ; (d) o) f > 34 I.E. Segal on -O. to the state space of A; (e) cJ > u) (U) on the state space of Q to the complex numbers. Now (c) is a Borel map, (b) Is easily seen to be continuous, and the other maps are obviously continuous. Therefore, if G is any closed set of complex numbers, /| /)^(U)G, o $ N J is a Borel subset of \ - ^/ , and as |\J is a Borel set, a Borel subset of I . Hence Lemma 4.2 applies and shows that J3 ^ = fcJ^ a.e. It follows that U),/ is extreme, i.e. pure, a.e., so that by [&] <f ] * is irreducible. PAHT II. APPLICATIONS 6. Decomposition oj[ a W*- algebra into factors. We show in this section that any W#-algebra with an identity (= rinp; in the sense of von Neumann) on a separable Hilbert space csn be decomposed into a kind of di- rect integral of factor?. A similar decomposition of a W*-algebra into more elementary W-a-alpebras is valid also for Inseparable spaces, but we are not then able to assert that these elementary algebras are factors. We begin by stating Just what is meant by such a decomposition. Definition 6.1. Let the Hilbert space /*f be the direct integral of the Hilbert spaces 7^ , as in Definition 4.1. An algebra Q of oper- P ators on ty is said to be the direct integral of algebras Q of oper- ators on 1f , with respect to the given decomposition of W as a direct P integral, if: a) every T d has a decomposition T(p) with T(p)e d a.e.; b) every decomposable operator T on /^ such that T(p) Q a.e. /> is in Q. . W e then write symbolically Q. = /ft dr(p). i/ p THEOREM 5. Let Q be a weakly closed self-adjoint algebra f operators on a Hilbert space fy , which contains the identity and has center C Then $ ll .. di rec t integral f factor?, relative t a decomposition f //" jas a strong direct integral whose algebra of diagonalizable operators is C . DECOMPOSITIONS OF OPERATOR ALGEBRAS. I 35 LEMMA. Q, contains a C-gubalgebra #" which jj* separable (in the uniform topology) , whose strong sequential closure Is Q , and which contains the Identity operator This lemma follows at once from von Neumann 1 s theorem that d contains a countable subset dense in the strong sequential topology [17 , p. 386.] PROOF OF THEOREM. Let S and be separable C*-algebras con- taining I which are dense in d and Q} respectively, in the strong sequential topology, and let 3^ be the C*-algebra generated by ) and C . Then j< is separable, for rational linear combinations of monomials in the elements of a dense subset of JD and a dense subset of & yield a countable dense subset of ^ . It is not difficult to see that CJ f =* # f , that * = $' = Q, -(by a well-known theorem of von Neumann), and that y = & * t = Q, n a = C. As C is abelian, C' contains every self-adjoint maximal abelian subalgebra of fl which contains C . Now such a maximal abelian suoalgebra is ki:ov/n to have a cyclic element, say z, which we can take to be normalized. A fortiori, C ! z is dense in A/-. Now ^ lf = C ! clearly, but *y f! is the strong closure of if 9 by the theorem of von heumanr. Just cited. Thus ^f is strongly uense in C 1 and it follows that *5^ z is dense in M We are now in a position to apply Theorems 2 and 3 with Q re- placed by !X. Let $/ toe the strong closure 01 fy(J3 ) . If T e & 9 then T is the strong limit of a sequence {T n } in f , but C c *? so that T is decomposable by Theorem 3. Putting r( ^ ) for its canonical decomposi- tion, so that a.e, T( / ) - strong lin^ T n ( /) if the sequence {T n "> is suitably chosen in J& , then for almost all Y 9 T( /) IP the strong limit of a sequence in Q^ 9 and herce is itself in d^ . On the other hand, if T is a decomposable operator with decomposition T( . ) such that a.e. T(/)e Q y , so that (Tx, y) = f('V( 1 )x( ^ ) , y( /)) dyx ( /) for all x 36 I. E. Segal and y in i^ 9 we shall show that T e Q If U e Q} 9 then by the argu- ment Just made U is decomposable with a canonical decomposition U(.) such that U( ^ ) is a.e. in the strong closure of *fj (& ) By Corollary 3.1, UT and TU are both decomposable, with decompositions U(.)T(.) and T(.)U(.) respectively. As each element of J& commutes with each element of & , each element of tjPj(f) also commutes with each element of 9^(<T). It lollows easily (using the identity of the strong and weak closures of SA algebras) that each element of the strong closure of $Mjff) commutes with each element of the strong closure of <pj()* In particular, a.e. U( /) and T( /) commute. It results that UT = TU, or T e Q M , i.e., Tea. Thus CL is the direct integral of the (2 / It remains to show that a.e. Q ^ is a factor. By Theorem 4, //* ^ is a.e. irreducible under 9^(7)- An equivalent way of stating this is as follows: ( <JVC30) f = ^^ a.e., where xJ/ is the algebra of all scalar operators on /Vy . Now *3* is generated as a C#-algebra by ft and , and it follows readily that f, (7) is likewise so generated by <f,(JP) and fy ( ) . It follows that ( 9^(T)) f = ( <?,(&))'*( ty,()). Putting 7*f, for the strong closure of 9?, ( G ) , as noted earlier ((2^) f = ( S^> ( ) ! > so ( CLf ) f n ( X^) ! = xJ^ a.e. Now each element of Q.^ commutes with each element of 7^ , i.e., ( ^)'^ Q^ , so ( Q^ )' n fl r c ( Q^ ) i "( ^) = J^. It follows that ( 0.^}* " Gij = J^ a.e., for as I e jtf, both (Q.^ and ( Q ^ ) ! contain ^^ , so that Q. ^ is a 1'actor a.e. ? Decomposition of a group representation into irreducible representations* We show next that every measurable unitary representation of a separable locally compact group is a kind of direct integral of irre- ducible continuous unitary reprerentations . This generalizes well-known results of Stcne and Ambrose concerning locally compact abellan groups (but its application to the special case yields a result which is considerably DECOMPOSITIONS OF OFBKATOh ALGEBnAS. I 37 less sharp than either that of Stone or that of Ambrose), and similarly generalizes a well-known analogous theorem for compact groups. In view of the known correspondence between positive definite functions on groupr and continuous unitary representations of groups [4], our result generalizes the representation theorem for positive definite functions on locally compact aoelian groups by showing that on a separable locally compact ^rroup, every measurable positive definite function can be represented as an integral of "elementary" positive definite function, where an "ele- mentary" function is defined as one which is not a nontrivlal convex linear combination of two other such functions (or alternatively, as one for which the associated group representation is irreducible)* A result closely resembling that presented in this section has been announced by F. Mautner [bj and is proved by him apparently witn the use of his result resembling our theorem on maximal decompositions (see Section 5), which we use in the following. Definition 71 Let U be a unitary representation of the topo- logical group G on a Hilbert space /V' We say that U is decomposed into irreducible representations by the (strong or weak) decomposition // c/ //' dr(p) if for every a*G, U(a) is decomposable, and if for p nearly all p e R, there is an irreducible unitary representation U of G such that U(a) is the decomposition of U(a). We then aa-y that U is the (strong or weak) direct integral of the U , or symbolically, U = yU dr(p). If G is locally compact, U is called measurable if (U(a)x, y) is measurable on G relative to Haar measure for all x and y in 7Y" (actu- ally, if G is separable such a representation is necessarily strongly continuous; cf. [l2j ) The regularity conditions which the method of proof of the follow- ing theorem could be used to establish are significantly stronger than those implied b} the theorem. In particular it is possible to define in a natural 58 I. E. Segal fashion, in case the representation has a cyclic vector, for all p in the perfect measure space M, a continuous unitary representation U p , such that [U (a)x(p), y(p)) is Jointly continuoas in p and a i'or x and y rang- ing over a certain dense subset of 7^, as well as with U irreducible i.e. and U = /U dr(p) The method of proof also yields a decomposition I'or strongly continuous representations of inseparable groups, in which :he constituents are irreducible in a kind of average sense (as in Lemma 4,2, but with continuous functions of 0. THEOREM 6, Every measurable unitary representation o^f a separable Locally compact group G is a direct integral of strongly continuous Irreducible unitary representations of G, Let U be a given strongly continuous unitary representation of } on the Kilbert space W. We put Ct for the collection (actually, o is is readily shown, a SA algebra; cf. [9] ) of all operators on "N* of the Torm /U(a)f(a)da with f eL-^G), where as in [9], yu(a)f(a)da designa- :es the operator which takes an arbitrary element x e/^- into the strong Integral J U(a)xf (a)da. (For a proof of the existence of this integral md for other facts concerning the operator thereby defined, cf. [9], and \Q] 9 esp. p* 83, and 84.) Let d be the uniform closure of d , so d is a C*-algebra. Now the mapping f > /U(a)f(a)da is continuous >n L^(G) to Q * anc * L^( G ) is separable, for the topological separabi- lity of G implies the separability of G as a measure space (relative to i regular measure) wnich in turn implies the separability of L^(G). It results that Q is separable, and hence Q is separable. o If z. is art arbitrary nonzero element of #" , the closure of z, is a closed linear manifold "/v' which is invariant under the U(a) [recalling that d is invariant unaer multiplication by U(a)) for a eG. [f z 2 is an arbitrary nonzero element in the orthogonal complement 1t ' DECOMPOSITIONS OF OPERATOR ALGEBRAS. I 39 of 1*t in ty , then the closure of Qz is a closed linear manifold Af (2) in W which is invariant under the U(a) and orthogonal to 7^ It follows readily by transfinite induction that there exists a collection ty ( ) ( ^ e ^) of closed linear subsnaces of // , mutaally orthogonal, with direct sum equal to //- , and each invariant under the ( ) U(a) and containing an element z/ such that Qz< is dense in 7V This shows that it is sufficient to consider the case in which Q is cyclic on If . *'or suppose the result has been established in this case. Then (I \ (t) t if U v * ' is the contraction of U to H , for each there is a measure space |x and decompositions ty ' = /#/ cljuj ^) anc * /" /) r ^ * / ^ V * c/ V diu (T^) Let | be the measure space whose set of U"T*1 T^| - A I f (we can and shall require that the |r are mutually 5 L, S 3 disjoint), in which a G'-finite measuraDle set is one which meets at most nT-l , and meets each |, in a measurable set, and in which the measure JJL of such a measurable set is the sum of the JUL^ -measures of its intersections with the |7 It is not difficult to verify that ^f is the direct integral of the /"/J 9 ver% ( I* /^)> the only condition which is not trivially verifiable being 2a) (note that as Q is separable, so is Qz for any z, so that the // are separable, and ( i } we can take the direct integrals of the 14^ to be strong). This follows from the fact that If z satisfies the condition in 2a), then (z(p), z(p)) * 0, except on a G'-finite set of p f s, say for pe ( J / / , and we can set z 1 5~ *!, where z\ is such that ^1=1 3 i i i i z'(p) z(p) a*e. for p e |j| , and z'(p) = for other values of p, the sum which defines z ! being convergent because (z f , z 1 ) * z'(p))dyu(p) = when i ^ J, and 5T || z^ || 2 j z f (p)|| ^/^f (p) s Atll z (p)ll dju(p) It is clear that nearly all the U^ * are irreducible and that U ** AiVS 40 I. E. Segal Suppose now that 6c has the normalized cyclic element z in fa^, and let C be a W-st-- algebra which is maximal abelian and self-adjoint in # f , C exists by Zorn's orinciple. We can now apply Theorems 1, 2, 3, ana 4. Utilizing the notations of these theorems, <jp ( Q] is a.e. irreducible. Now every (uniformly) continuous self-adjoint representation *f of CL induces a unique continuous unitary representation V of G such that <p ( /U(a)f (a)da) = yv(a)f(a)da for f 61^(0) and with the property that <f is irreducible if and only if V is (see (8] and [9j), If we put U^ for the representation of G induced by ^ , it follows that U^ is irreducible a.e. and that U = /Uj d u( /) . 8 Decomposition of an invariant measure into ergodic parts* We show in this section that a regular measure on a compact metric space which is invariant under a group of homeomorphisms can be represented as a kind of direct integral of ergodic measures on the space. We recall that an ergodic measure is one relative to which every invariant measurable set is either of measure zero or has complement of measure zero. Definition 8.1+ A finite measure m on a o'-ring * is said to be a direct integral of measures m , where p rarges over the set H of a measure space M = (R, 1Q 9 r), if for each peR m is a finite measure on f , ana if also for each E e J 9 m^E) is integrable on M and ^/ p (E)dr(p) = m(E) . The proof of the 1'ollowing theorem yielas a kind of maximal decom- position of invariant measures into invariant submeasur'es in the inseparable (compact) as well as separ^ole (compact metric) case, but we are unable at present to establish ergodicity of the submeasures except in the separable case. A number of similar decompositions have been obtained by quite different methods, for the cases of one-parameter and infinite cyclic groups, the first such result being due to von Neumann, and the most general one being that in 2a] which applies to a class of separable measure spaces DECOMPOSITIONS OF OPERATOR AIjGEBRAS. I 41 including the one considered here. THEOREM 7. A regular measure o_n a. compact metric space M which l invariant under a group G of homeomorphigms o_f M _i_s a d! rect Integral o G-ergodic regular measures on_ M. There is clearly no loss of generality in assuming that m(M) 1, where m is the measure in question. We call a bounded measurable func- tion f on M invariant if f(a(x)) = f(x) a.e. on M 1'or all aeG. Let W" be the Hilbert space ^f all complex-vyl"ru functions square- integrable relative to m, the inner product of two elements f and g of /-/ being defined as / f (x) < g(x)dm(x) . Let Q be the algebra of ail ^M operators a on // of the form f(x) > k(x)f(x), f 6L 2 (M, m), where k is a continuous complex-valued function on M, and let C be the algebra of all Q, with k complex-valued, bounded, measurable, and K. invariant. We show next that CL is separ-*>ble in the uniform topology and that C is a W*- algebra. Let \N. ; 1=1, 2, } be a basis fur the open sets in M, and let {?i n > n = 1, 2, ...J be for each i a sequence of continuous func- tions on M which are uniformly bounded and converge ( 'icir.twi se) to the characteristic function of N>. (E.g., if -fc. "| is a monotone increasing seauence of closed subsets of N. such that N* = \l C , then f. can 1 - 1 n in in be taken to be a continuous function with values in [p, ij which is 1 on C in and outside of N., such a function existing by Ur}sohn ? s lemma and the normality of a compact hausdori'f space). Then the rational linear combinations of the f ln are dense in C(M). i*'or otherwise, 0} the Eahn-Danach theorem there would exist a nonzero continuous linear functional CT> on C (M) which vanishes on the ^i n N w lt ^ known that every such functional has the form <p(f) = / f(x)dn(x), for some regular countably- M additive set function n on M. It follows easily that 42 I. E. Segal / N dn(x) n(N.), so that n vanishes on all the N. It is not difficult to show that any finite union of the N. differs by a set on which Var n is arbitrarily small from a finite disjoint union of N^s, which implies that n vanishes on all finite unions of the N., and hence on all open sets, and so by regularity vanishes identically Thus the f, are fundamental in C(M). It follows that C(M) is separable, and as the map k > Q^ is continuous on C(M) to the operators on ty in the uniform topology (in fact ||k||i || Q. ( ), the image Q. must be separable. For a e G, let U a be the operator on ~H* defined by the equation (U ft f)(x) = f(a(x)), f e //. Then it is easily seen that a bounded measurable complex- valued function k on M is invariant if and only if U a Qic = QU for all aeO. It follows that if ty is the algebra of all a with bounded measurable k, then C = ^ n [U[aea] f . Now it is known [15] that >Y a ^ f , so that in particular H^ is weakly closed, and as it is easily verified that Qr (\)*> ty ls s * and so a W*-algebra. Plainly, ("U | aeO J 1 ia weakly closed, and it is easily seen that U a a is unitary, so that (^ a )* = U showing that [ U & | a e G ] is SA. a"^ Hence [ U a | a e G ] f is a V\i*-algebra. Thus C is a W*-algebra containing the identity. Now let z be the function which is Identically unity on M. Then Oiz consists of all continuous functions on M, and so, by virtue of the regularity of in, Cti is dense in / . It follows that the conditions of Theorem 1 are satisfied, and hence for any T 6 d and S C , (TSZ, z) = with | , O)^ , and JLC as in Theorem 1. Now the states of d are well known; for each u)y there is a regular measure m^ on M such that r o)^(Q k ) * yk(x)dny (x) for all keC(M). We show next that for almost all tf , m is an invariant G-ergodic measure. DECOMPOSITIONS OP OPERATOR ALGEBRAS. I 43 It is easily seen that ty^U -i m % > so that U a TU a -l e & if Te&, and (U a TU ft -ia, z) J^(^) ^/U a TU a -l)d/x( /) On the other hand, if T = Q k and S Q with T e Q and S C, then (TSz, z) k(x)p(x)dm(x) and (U a TU a .iSz, z) = yk(a(x))p(x) dm(x) (for m and p are invariant) = (TSz, z). It results that J&( aO 60^(U a TU a -l)du( /) = J&Cf) *>f(T) dyu(ef). It follows from the arbitrary character of S (every bounded measurable function on | being an S(.)) that oV(U a TU _^) = cO^(T) for almost all /, and since both sides represent continuous functions, the equality for all if follows. But if T Q^ o)^(T) ^(xjdm^ (x) and ^^^^ - ^(afx) )dny(x), so for all continuous functions k on M, /k(x)dm^/ (x) ^ktatx) )dm^(x) It follows that m/ is invariant for all if . We have for any k e C(M), putting T ^ and S I in the above formula, that /k(x)dm(x) = / [ / k(x)dxn^ (x) J dyu( /) . Now if ^ is the set of all bounded Baire functions k on M for which this equation holds, it is easily seen that /^ is closed under bounded point- wise convergence, and as fy contains all continuous functions, it consists of all bounded Baire functions. Now if E is any Borel measurable set, r> its characteristic function is Baire and so m(E) = /,m^(B) d u ( / ) . Thus m is the direct Integral of the ny over (] f JJL ) It remains only to show that my is a.e. ergodic (in fact the preceding decomposition of m into invariant sub-measures is valid without the separability assumption on M) Now the ergodic invariant regular proba- bility measures on a compact space are precisely the extreme points of the set of all invariant regular probability measures on the space (the proof of this in [9j is for the group of reals under addition, but applies to an arbitrary group with trivial modifications) The set of invariant regular probability measures n on M is also known to be in one-to-one convex linear correspondence with the set JT of all invariant states V on d 44 I. Segal ( T/> being invariant if 1>(U TU .1) = v(T) for TeA and aeG), where a a n and v correspond if V (Q^ * yk(x)dn(x) for all keC(M). Now ^~ is a convex set which is compact in the weak topology (recalling that the state space of a C*- algebra with an identity is compact)* Moreover, j~ is metrlzable, for if XT. (i = 1, 2, ...)} is a countable dense sub- set of & , with no T = 0, the metric d(n, p) = "" 2" 1 || T H" 1 |n(T 1 ) - p(T ) | is easily seen to induce a topology on ~~ identical with the weak topology. It follows, by an argument used in the proof of Theorem 4, that if m^ is not ergodic a.e., then there exist for each "3^ invariant stat Jd f and Cy such that 1) V = (1/2) ( yO, + <r r ) for all / ; 2) if T (2, /^( T ) and V( rr ) aro Borel functions on | ; 3) yO. ^ Cy for a measurable set of if f s of positive measure. As before, there exists for each if an operator S^ in ( ^(A)) f such that yO/(T) = , z(/)) and || S^ || f 2. Now (S, ^ (X) ' ^( y )) = ), z(/)) =yO / (Y*X), and is a measurable function of / . It follows that (Sy x(^), y(y')) is a measurable function of if for x and y in ^ , and this function is integrable for |(Sy x( / ) , y(/))| f 2|| x( / ) || || y( ^ ) || , which bound is integrable by Schwarz 1 inequality. Hence there is a decomposable operator S on 7^ for which /Sy' d/* ( ^ ) is a decomposition. By Theorem 2, S e ft 1 . We show next that fl ! = )^, so that S is a multiplication by a bounded measurable function on M. Now if |f 1 is a sequence of bounded measurable functions on M which are uniformly bounded and which converge a.e. to a function f. then fy. con- 1 n verges weakly to Q^, by a simple computation. On any finite regular measure space, every bounded measurable function is a limit a.e. of a bounded sequence of continuous functions. It follows that the weak closure Q. w of Q contains ^ . Hence Q W 'C: ty 1 , but /2 W = <2% so # w f ft 1 , and O'er Jf 7^. DECOMPOSITIONS OF OPERATOR ALGEBRAS. I 45 Thus S ~tf[ and so S = Q k . Moreover, S 6 C, for if a is arbitrary in and T arbitrary in Q , then by the invariance of ft ^ , (S, T^V^a-l'** *^' z( ^ }) = <S^ ^(T)z( ^), z(t)). Integration over | of this equation shows that (SU a TU a iz, z) = (STz, z) (noting that (U a TU & . 1 z)( ^) = ^(U a TU a -l)z(^ ), by Corollary 2.1 ). If S = Q k and T = Qp, this means that yk(x)p(a(x) )dm(x) = yk(x)p(x)dm(x) , which implies that ykla'^x) )p(x)dm(x) = yk(x)p(x)dm(x), from which it follows readily that k(a" 1 (x)) = k(x) a.e. for each a e G. That is, Se C, so S/ is a.e. a scalar multiple of the identity, and yO^ = cr/ = <^ a.e., a contradiction. Hence m^ is a.e. ergodic. 9. The Fourier transform for separable unimodular group^. We show in the present section how the Fourier transform as defined in D--U can be correlated with the Fourier transform as an integral whose kernel is an irreducible group representation. If f is an integrable function on the locally compact abelian group G, its Courier transform is usually defined as the function F on the character group G* of G, defined by the equation F(x*) = / x*(x)f(x)dx. The generalized (Weil-Krein) <s r\ r 2 r 2 Plancherel theorem then asserts that / |F(x#) | dx* = / |f(x)| dx, ^ G* ^G for f 6Lo(G) The Fourier transform can be extended to compact (not necessarily abelian) groups by replacing G-* by the collection of continuous irreducible unitary representations of G (which is simply the character group when G is abelian); one has then F(/>) = / /)(x)f(x)dx and the generalized Plancherel theorem (usually called the Peter-Weyl theorem in this context) asserts that J |f (x) | 2 dx = ]>~ tr( (F(^ ) )*F(y2 ) )d(yO ), where d( /D ) is the degree of ft , tr denotes the usual trace, and the sum is over any collection of representatives of equivalence classes of irreducible representations of G. In the case of an arbitrary separable unimodular group, it turns out that the same formal relations are valid, provided "irreducible 46 I. E. Segal unitary representation" is replaced by "two-sided irreducible unitary representation 11 . As indicated In (jLOj, this does not materially affect the situation in groups which are either compact or abelian* More specifically, in [llj it is shown that if the Fourier transform is defined through the use of the von Neumann reduction theory, then the Plancherel formula for a separable unimodular group holds, the trace now being that defined by Murray and von Neumann for factors, and the integration being over a measure- theoretic analog of G*. As it can be verified that the reduction obtained in Theorem 2 satisfies von Neumann 1 s conditions (cf. the last theorem in this paper), the Plancherel transform P( tf ) of a function f e 1^(0) A L^fG) can also be dei'ined through the use of the present decomposition theory. We shall show in this section that the Plancherel transform of f can also be obtained as follows: i'or each ^ e |"~^ , where ( | , LL) is the perfect measure space on which the decomposition is built, there is a two-sided continuous unitary representation [L ^ , R^j , where L^ and Hf are respectively the left and right ordinary representations of which the two-sided representation is composed, which is a.e. irreducible, and such that F( ^ ) = f L, (a)f (a)da. ^0 * We begin by considering the decomposition of conjugations in suitable situations. We recall that a function J on a Hilbert space /*J^ to itself is called a conjugation if it satisfies the conditions: 1) J 2 = I, 2) (Jx, Jy) = (y, x) for all x and y in // ; and that it has the properties J(x + y) = Jx + Jy, and J(OLx) = 2x for all x and y in 7/ and complex OL It follows that the map W > JWJ is a ring automorphism of the set of all operators on // , and that (JWJ)* JW*J, We denote JWJ by W J , and designate the automorphism W > W J by J. THEOREM 8. With the notation of Theorem 1, let J be a 'conjuga- tion of /y. such that S J s S* for all S e C , and Jz = z . Then there exists for each / e | a conjugation J^ on /^ such that for any DECOMPOSITIONS OF OPERATOR ALGEBRAS. I 47 and y In H , (Jx, y) ^i *< *>* y( We first define fy on >f^U) as follows: J^ >^( To see that this definition is single-valued, observe that if ?^(W), then ( ^(T) - ^(Vk) f ^,(D - ^(W)) 0, so that cJ((T - W)*(T - YO ) - 0. Now u) is transformed into u) by (J)* so that u) (J(T - fl)*(T - W)J) = = ^((J(T - W)J)*(J(T - VY)J)) - >^(W J ), >/ r (T J ) - >f,(W J )), so that We show next that for all if and for T e - >f,(W J )), so that ?^(T J ) n/(W J ) Let S be an arbitrary self-adjoint operator in C . Then u> (ST ) yS(^) o) | ,(T J )d/>l( / ). On the other hand, cJ (ST J ) tJ(S J T J ) (for a SA element of C is invariant under J) = u)((ST) J ) cj(ST) ) dJ / (T)d > u(^). As S = S, it results that y S( / ) {^/(T J ) - dyu( 3^ ) = 0. As S(.) can then be an arbitrary real- valued bounded measurable function on / , it results that cJ^(T ) - t*V(T) * a.e., and since the left side is a continuous 1'unction on I , wo have Now for any T and W in Q we have (J^ y,(T) , J, 7 r (W)) >( / (W)). In particular, || J^/T)|| ^ || ^(T) || , so that J^ is bounded on >I^(CL) 9 and therefore has a unique continuous extension to the closure 7 of >($); we denote this extension also by jy Prom the equation (J r >^(T), J, ^(W)) = ( >^(T), /() it follows by continuity that for arbitrary x^ and y^ in ^ , (J r x^ , J^ y^ ) y^ ). Now J? ^(T) * J, (J, >?,(T)) * J^ >f,(T J ) = >f r (T), so that Jy is the identity on >?/(#), and hence also, by continuity, on ^/y Thus J/ is a conjugation. It remains to show that for arbitrary x and y in ty , (Jx, y) = c/p (J, x( ^J, y( 3f"))dyu( /). Now if x * Tz and y Wz with 48 I. B. Segal T and W in <2 , (Jx, y) = (JTz, Wz) = (W*JTz, z) = (W*T J z, z) (using the fact that Jz = z) = o /J^(W*T J )d^( / ) * J"( ^(T J ), ^ (W) )dyu.( /) = J ( J * ^/^^ >fy(W))dyu.( a( ) TJaus the equation is valid for a dense set of x and y f s, and it follows as in tho first part of the paper that it is valid for all x and y in "^, When uJ^ is not a state, J^ can of course be defined arbitrarily. The next theorem asserts that the conditions of the preceding theorem are satisfied (with a suitable choice for z) in the case of a certain conjugation on L (G) for a unimodular group G. 2 THEOREM 9. Let G be a separable unimodular locally compact ftroup, and let Q b the C- algebra generated b^ all operators on 1^f = L P (G) of the form L~R , with f and g in L, (G), where L*, and R_ * ~"~" - r g i i g are respectively left convolution by f and right convolution by g Let C be the center of the weak closure o Q Then if J is the conjuga- tion on ^/ defined b the equation Jf = f* for f e #; where f*(x) = 7(x" ), every self-adjoint element of C JL invariant under the auto- morphism induced bj J (i JSJ = S* j.f S C ) ; there exists an element z jLn 'ty such that Ci z _is dense jg ^ and Jz = z ; and Q f - ^ We show to begin with that C = Q\ and that the weak closure of CL is the V\i*-algebra generated by L and R, where L and R are respec- tively the ^*-algebras generated by the L and R , these operators be- a a ing defined by the equations (L a f)(x) fta" 1 ^) and (R & f)(x) == f(xa), f e "# and afeG. Clearly Tea 1 if and only if TL f R g = L f R g T fol% all f and g in L (G), i.e.^if TL f R h = L R Th for such f and g and all he #-. Now if g } is a sequence in L,(G) such that h * g^^ > h and (Th) * g n > Th, it results from the equation TL f Rg n h L f Rg Th that TL h = L f Th. Hence TL f = L f T, and as -^ is generated by the L f (cf. [10]), this shows that T ' . Similarly T e ^t 9 and so we DECOMPOSITIONS OP OPERATOR ALGEBRAS. I 49 have Q f c^-#. it is shown in [lOj that . ft, so a'c/?^ 1 , which shows that Q. 1 is abelian. It follows that the center of Q n is f , and as Q_ is SA, its weak closure is <2 H , so that C * fl f On the other hand, L f e <L and R f e / for all feL-jO) (loc. cit.), which implies that d c "ft. Hence Q f is/ 1 ^ #' and it follows that a 1 = n # or a" =a ^ /?. Next we show that for an arbitrary element S of C , S S*. The mapping T > T J is easily seen to be continuous in the strong operator topology. As the finite linear combinations of projections in O are strongly dense in C , it therefore suffices to prove S s S* r^t for the case when S is such a linear combination. As J is conjugate linear, it is enough tc show that P = P for every projection P in C Now let 7^ be the range of such a projection P. It is shown in [ll] that "7^ is then a two-sided ideal in the Lg- system of in the sense of Ambrose [2] . According to a theorem of Ambrose (loc. cit. Th. 7), every such ideal is invariant under J. Hence for x 7=/, JPx 6 7^ f i.e., P(JPx) = JPx, or PJP = JP* Multiplying the last equation on the left by J shows that P J P P. Now P is SA, so (P J P) * P and PP J ? /x/ J J Applying J to both sides of the last equation shows that P P * P , so P - P J . It remains to show that there exists an element z of H such that Q.Z is dense in ty and Jz = z. Let {z^} be a family of elements of /^ which is maximal with respect to the properties 1) Jz^ * z^ and z. ^ 0, 2) Ctz. is orthogonal to Qz, if i ^ J. Then the index set over which i ranges is at most countable, for the closures of the Qz^ constitute a fa-nily of mutually orthogonal closed linear subspaces of & , which by the separability of # must be at most countable. We assume that i = 1, 2, ... and put z =JEI n " II z II *& Plainly Jz z, and we show finally that Qz is dense in ^* 50 I. E. Segal Assume on the contrary that dz is not dense in /T^* Then there exists a nonzero element x in // which is orthogonal to Ctz. Let ^f^ be the closure of Qz. and P the projection operator on /V' with range 7^ . We have (Tz, x) * for T e d , so (S*Tz, x) = for S and T in d , or (Tz, Sx) = for such S and T. It is easy to deduce that the last equation is valid for all S and T in the weak closure of CL . Now it is easily verified that ^ is invariant under & , so that P # f - C, and hence P e <2 M . Hence we have in particular, for T e d, (TP 1 z, PjX) = or (Tz^ P^) = 0. Now P^ ^ but the Tz span 7^4* so P 4 X = * As x ^ 0, I -^ P^ = Q is a nonzero projection in C Putting w for any nonzero element of Q & such that Jw ^ - w (obviously such a vector exists) and z 1 = w + Jw, clearly Jz f = z 1 and flz' = flQz 1 = Q<jUCQ>f, so that Qz f is orthogonal to all the Qz^ 9 contradicting the maximality of x 2 ^}* Before proving the result mentioned in the beginning of this sec- tion we make appropriate definitions. Definition 9.1 A two-sided representation f a group G on a Hilbert space "^ is a pair (L, R) of one-sided representations of G on >/ such that L(a)R(b) = R(b)L(a) for all a and b in G. If G is topological, such a representation is called strongly or weakly contin- uous if both L and R are (respectively) strongly or weakly continuous* If (L, R) is a two-sided representation of G on W 9 if each of L and R is unitary, and if J is a conjugation on 1*fr such that JL(a)J * R(a) for all aeG, then the system (L, R, J) is called a two-sided unitary representation of G* A two-sided representation (L, R) is called irre - ducible if the L(a) and R(b) (a, bG) leave no closed linear subspace of 7-f (Jointly) invariant other than and ft . The following theorem shows the connection between the Fourier tram form as defined directly thru reduction theory and as defined thru the use DECOMPOSITIONS OF OPERATOR ALGEBRAS. I 51 of an integral whose kernel is a representation* THEOREM 10. Let G, 14 9 d C , J, and z b ajj in the preceding theorem, and let ]~\ iu , and u) ^ be a in Theorem 1* For every f L. ( G ) , the left and right convolution operators L and R are decom- posable with respect t the reduction of ~W described ijn Theorem 2, with M = ( P, yu ) For almost all / e P there ^s_ a two-sided strongly con- tinuous unitary irreducible representation ^L ^ , R ^ , J y j- of G on 1*4* ^ which i almost everywhere on ( P, ^u ) irreducible, and such that the decompositions of L and R are / L^ (a)f(a)da and J R^(a)f(a)da respectively^ We observe to begin with that for any aeG and Xe OL 9 L X and R X are in OL * For let Q be the algebra generated by the L f E a with a o x o f and g in L (G); as every L commutes with every R this algebra consists simply of all finite sums of such operators. Now it is easily veri- fied by direct computation that L L f = L f and R & R R a , where f ft (x) f(a -1 x) and g a (x) g(xa)* It follows that fl is invariant under left multiplication by the L and the R , and it is not difficult a a to deduce by an approximation argument that so also is OL invariant (cf. [8], p. 80). A trivial modification of a proof in loc. cit. shows also that R X and L X are continuous functions on G to d , in the uniform topology on Q The remainder of the procedure for obtaining the L^ and the R^ is also similar to that used in loc. cit., and we shall merely outline it* We define L^ (a) on "^/(A) by the equation L^ (a) >ff(T) = ^(L R T). It is easily verified that L^(a) ^(T) is single-valued and that for each aG, L^ (a) is an isometry on T/^Cfl)* It can therefore be unique- ly extended to an isometry, denoted by L^(a), of "/"/y into ^>. The mapping a > L^(a) is a representation, and hence so is the mapping 52 I. E. Segal a > Ly(a). Plainly L ^(e) - L^ (where e is the group identity and 1^ the identity operator on 7V^), so L ^ (a)L ^ (a" 1 ) = L ,/ (a -1 )L ^ (a) = lj , which shows that L^(a) is unitary. Now the map a > L a X is con- tinuous on G to 0. 9 and "^ is continuous on 0. to /V^ , so L^(a) >?y(X) is a continuous function on G to 1^ , for any fixed Xe#. It follows by continuity that L^(a)x^ is continuous as a function on G to 7^ for each fixed x^ A*V. Similarly for the definition and properties oi' R^ . (These definitions are for the ^ such that u} ^ is a state; for the null set of other ^ we take L^ and R^ to be the identity representation and J^ to be an arbitrary conjugation on Hy Now L^X = E^L X for any X Q., which implies that Ljf (a)R^ (b) ^(X) = R ^(b)L^ (a) >^(X}, and as L ^(a) and R ,/(b} are bounded and ?t^(Q) is dense in 7/y , it follows that L y (a)R ^ (b)Xj/ = Rf (b)L^ (a)x^ for all x^ e /=/y , I.e., L ^(a) and R^(b) commute for all a and b in G. Next, it is easily verified that JL ft J =* R , and it follows that for Xe #, JL & XJ = R JXJ, or (L a X) J = R & X J . Hence ^((L X) J ) = n^R xJ )t and b y the definition of J^ in the proof of Theorem 8, J^ ^(L ft X) R^ (a) ^(X J ), or J^ L^ (a) ^(X) R/ta)^ ^(X). Thus the bounded linear operators L ^(a) and J^ R^ (a)J^ agree on the dense set yf^(Q.) and therefore coincide. Now if f and g are arbitrary in L (G) and X is arbitrary in Q, , we have ^ (L f R ) >j^(X) = ^(L f RX). We note that r* p ^6 ^5 t //L R.X f(a)g(b)dadb exists aa a strong vector-valued integral (i.e., relative to the uniform topology on d ), and equals LfRJt (<^ loc. cit.)< AS //- As 1(f is a continuous linear operation, it results that Y / (L a R.X) f (a)g(b)dadb, but fy(L RJO = L^ (a)R^(b) f^(X), so we have /*/* 9^(L f R )x y = J Jl>s (a)R r - (b)x^ f(a)g(b)da db for x^ s Y/(^) A S >f^ ( tf ) is dense in 1^ f , and as T^/'^f^g^ and (a)R^ (b)f(a)g(b)dadb are bounded linear operators, it follows that DECOMPOSITIONS OF OPERATOR ALGEBRAS. I 53 the preceding equation is valid for all x^, e /Vy. Now if ^g } is a se- quence in 1^(0) such that g^a) - 0, ^L(a)da = 1, and g vanishes outside of W , where D W n = e i then Lj,Rg is easily seen to con- verge strongly to L~, so that L is decomposable, and similarly R ia If 5 decomposable. On the other hand, by the Fubini theorem for vector Integra- tlon yy L /( a > R i' &>** f(a)g n (b)dadb ^y^y (b)Cyii^ (a)x, f (a)da]g n (b)db, which expression is easily seen, by virtue of the strong continuity of R^ , to converge strongly as n > co , to </L j, (a)x^ f (a)da Thus the decomposition of L* is as stated, and similarly for that of R It remains only to show the irreducibility a.e. of W^ under the combined action of the L ^ (a) and the R^(b), a and b being arbitrary in G. Now if a closed linear manifold in 7"/y is invariant under the L^(a) and the R ^ (b) it is also invariant under 9V< L f R g ) for all f and g in L,(G) (cf. loc. cit.), and hence is invariant under *f * ( & ) Now as shown at the end of the proof of the preceding theorem, C - ( Q ) ! , where Q is the weak closure of d , and it follows that = d* This shows that ti is maximal abelian in Q} . The separability of L.(G) together with the continuity of the maps f > L^ and f > R. on L (G) to d 9 implies the separability of # o , from which the separability of (2 follows. Hence Theorem 4 implies that ^(0.) is Irreducible a.e, Remark 9.1+ In the special case (for semi-simple Lie groups, con- Jecturally the general case) that ^ is a direct integral of factors of type I, the situation can be further reduced, in that the corresponding two- sided irreducible representations of the group G arise in an obvious fash- ion from one-sided representations. Specifically, for almost all tf , there is a Hilbert space K^ , a strongly continuous irreducible (one-sided) representation U^ of G on Yt^ , and a conjugation Cy of K^ > such that the foregoing two-sided representation -^L^ , R, , J r J is unitarily equivalent to the representation {L^ , R ! , , J^} of G on the Kronecker 54 I- E. Segal product 14\ ** ~tf # 1{ > where LJ, , R^ , and jy, are determined by the equations L', (a)(x#y) = (U r (a)x)#y, RV(a)(x#y) x#(C, U^ (a)<V y), and JV (*#?) * (r y# c * x) > for ftl1 * and y in /t^ and aeG. If % ^ is taken as Iv>(H) for a measure space M = (R, $, r), then fify can be taken as Lg(MXM) and J^ can then be defined by the equation UJ- f )(x, y) f(y, x), f e * * To see this, let <^ and i?^ be respectively the W*-algebras generated by the L^Ca) and the R^(a), aeG, and observe that <^ ). For if S zdx( ^), sa y SS r dx(^), then S e for all ^ Now if W R f with f 6^(0), then W where W^ yR^ (a)f(a)da, and it can be seen that W^ e R^ (cf. Hence S^ and W, commute, and it follows that S and W commute, n which implies Se{R f } f or S e -tf <. Thus "=>J<^^^( /). On the other hand, it follows from [10] that every element of Ju is a strong sequential limit of (bounded) operators of the form L f , with feLg(G), Every such operator in turn is a weak sequential limit of operators of the form L f with fl,(G), for if h and g are arbitrary in LAG), the integral Jt (y)g(y*^x)dy exists and by virtue of the boundedness of L^, equals (L f g)(x) (see [ll]), and by Pubini's theorem the integral J Jt (y)g(y"^x)h(x) dxdy exists. It follows from the Lebesgue convergence theorem that if (K } is a sequence of compact subsets of G such that Kj^oK^ and f vanishes nearly everywhere outside of LJ K D * and if f n is the product of f with the characteristic function of K^, then JLf j converges weakly to Lf As weak and strong sequential limits of decomposable operators are likewise decomposable, it follows that every operator in L is decomposable, i.e., C /-^^dyu( /), and hence jL * Now by the irreducibillty a.e. of l^, R^ , J^ we have a.e. * 6^, where $^ is the algebra of all operators on 1^ Clearly DECOMPOSITIONS OF OPExiATO* ALGSBfaAS. I 55 f .3 ^ It follows readily as in the proof of Theorem 5 that <> ^ and fj, are factors. Now assuming that ^ Is of type I, we shall show that , R r , J^j- has the special form given above. For this It Is evident* ly sufficient to establish the following lemma, which includes a result recently announced by Oodement 4a ], LEMMA. Let -JL, R, JJ- be a two-sided Irreducible strongly con* tlnuous unitary representation of a topo logical group on a Hilbert space 7V-, and let the W-algebra generated b the L(a), a eO, be of type ! Then there exists a one-sided Irreducible strongly continuous representation U f on a Hllbert space ^, and a conjugation C of "W 9 8uch that {L, R, J, 1+} jji unltarily equivalent to the system {L 1 , R ! , J 1 , 7-/ 1 } where ft* * K#K> J ! Is the conjugation of /y 1 de- flned bjr the equation J f (x#y) * Cy#Cx, for all x and y jln K , and o^ ' and ~fc* are the representations of defined by the equations L'(a)(x#y) =* (U(a)x)^ and R'(a)(x#y) (x#CU(a)Cy) for all x and y in ^. By [7 , pp. 138-9 and 174], H is unltarily equivalent to TV 1 for suitable Hilbert spaces and K, in such a way that is mapped into the set Z? of all operators of the form S#I with 3 an operator on "H , and the W-algebra /f generated by the R(a) is mapped into the set 7?. of all operators of the form I#T with T an operator on # Now J maps into a conjugation J 1 of /V 1 with the property that J'^.J 1 /? . As the dimension of a Hilbert space is the maximal number of mutually orthogonal minimal projections in the algebra of all operators on the space, and as the mapping X > J'XJ 1 la a ring iso- morphism preserving ad Joints, ft and # have the same dimension and we can set ft^ K 2 ft. Plainly L(a) is mapped by the foregoing equivalence into a unitary operator L f (a) on %# 7\, of tha form 56 I. E. Segal L lf (a)#I, where the W*-algebra generated by the L^a), a e G, is ,. Now it is easily seen that the map T#I - > T from . to the operators on A^ is strongly continuous. It follows readily that the map a - > L rt (a) is a strongly continuous unitary representation of G on 1"C , and that the strong closure of the algebra generated by the L lf (a) is the algebra ~tS of all operators on ft. The latter feature implies that L !l is an irreducible representation. Similarly R(a) is mapped by the foregoing equivalence into I#R w (a), where R rt is an irreducible strongly continuous irreducible unitary representation of G on 1^ . For any T e & we have clearly J ! (Tl)J f = I# ^(T), for some func- tion <p on & . It is readily verified that ^ is an (adjoint-preserving) ring automorphism of 15 of period 2, and with the property that <jP(QLT) = C6<jP(T) for complex <X and T 73 . It is not difficult to deduce that there exists a conjugation J fl of ~K such that <f (T) = J n TJ" for T e 73 . Now let C f be the conjugation of W determined by the equation C f (x:#y) = J l! y#J fl x> for * and y in A. It is easily seen that J T C' is a unitary op- erator U 1 on H f with the property that U 1 *.^!! 1 = ^. As A is algeb- raically isomorphic to 13 9 every automorphism is inner, and so there exists a unitary operator V 1 in such that U f *T'U' = V's-TW for all T < It follows from the last equation that U'V'" 1 * so that U'=V ! W with a unitary operator in 7^. Evidently V = Vl and W f = I#W, where V and W are unitary operators on *H 9 and it results that J f Now J'(Tl)(x#y) = (I# <f (TDJ'Ufry) for all x and y in K, and sub- stituting the above expressions for J f and for <$ , it is found that (VJ fl y)#(WJ ff Tx) = (VJ tl y)#(J M TJ"VYJ ll x). It results that V<J n T = J^TJ^J 11 , and multiplying on the left by J", it follows that J"WJ" commutes* with T. As < is arbitrary in V, this implies that J"Vi/J fl = I and hence W = I. The * 2 o j , J 1 =1 implies that J f (x^y) = x?jry for all x and y in ft, and DECOMPOSITIONS OF OPERATOR ALGEBRAS. I 57 substituting J' = (V#I)C there results the equation ( Vx)iKT"VJ"y) = x#y. Hence V = I and J 1 = C. The proof of the lemma is concluded by the observ- ation that as JL(a)J = it(a), a e G, we have K"(A) = ^(L"(a)) so that h tt (a) = J"L"(a)J". 10 - Deflation of decompositions. In this section we consider the problem of replacing the regular measure space ( P* , JJL ) whj ch has figured in the preceding decompositions by spaces which are measure-theoretically equivalent, but which have different topological properties. Our first result asserts roughly that under appropriate, but fairly general, circum- stances, ( I , JJL ) may be replaced by a regular measure space ( A , V ), which is a kind of "deflation" of ( V , yu ) which arises naturally in certain circumstances. For example, in the case of the reduction of the regular representation of a locally compact separable abelian group G, the measure ring of ( I , yu ) is identical with that of the character group G* of G under Haar measure; but I is topologlcally much "larger" than G*, roughly speaking. The general process described in the next theorem could be used to replace I in this situation by G*. THEOREM 11. Let Q , C> and be a in Theorem JL, and suppose Q, con- tains the identity operator. Let C b the closure ija the uniform topology the algebra generated b all functions on P o the form cJ^(T), with T d . Let ^\ be_ the (unique) compact Hausdorff space such that i iaomorphic o C( A ) 9 and let ^ be the continuous map o_f P onto A such that ^f f , and if f corresponds t F e C ( A ) in. the isomorphism f with C( A ) , then f ( / ) = F( <? ( f ) ) for all ^ e P Then setting Tj = 60^ when <5= <p(f) , there j^ ,a regular measure V p_n ^ such that ^) o^^(T) d>u( /) ^t 6 (T) dv( 6) ; (2) b^ t G> 2 implies that T-& ^ T-S ' ^' ^ ne Capping (3 > Yb i continuous on_ ^ ^ the state space o_f d ; (4) j. the hypothesis f Theorem ^ Is satisfied, then 58 I* E. Segal t& ll pure for almost all b relative to ( A , V )5 (5) 1 Q l dense iH C f IS ttie weak sequential topology, then there i an algebraic isomorph- ism S > S 1 ( . ) of C onto the algebra of complex- valued bounded measurable functions on ( A , V ) such that (STz, z) = J^ S 1 ( O ) t^(T) d v( c> ) for all S C and T e & . The existence of the A and <f described in the theorem is assured by known results [ 14 J . We define V on Borel subsets E of A t>7 the equation: V(E) = ^( <p "^(E) ) ; then it is readily seen that V is a regular measure on &. If f is an arbitrary real-valued element of o , then cf( ^)dyu( f) = jf^*pl /| f( f ) < X ] - \tvl c)|F(>) </\J, where F( ? ( ^ ) ) = f ( a" ), F being in C( A ), and so <y/f( ?T )dyu( ^ ) = /F( O )dv( c) ). Now defining T& as above, (1) and (3) are obvious, and (2) follows readily from the fact that f ( ^) = <f ( X ^ if and only if f( /^) = f( ^ ) for all f S . To see (4), observe that the inverse image under <? of the Borel set [ 6 | t^ is not pure] is the set [ ^ | a)f is not pure J , which by Theorem 4 has measure zero* To establish (5) we need the following lemma. Our method of proof here could be used to give a simple demonstration of a theorem of Dieudonne 3 ] . LEMMA 11.1. Let ( JP , yu. ) be a compact perfect measure space and o a uniformly closed SA subalgebra of C( P ) which jj^ dense in C( T* ) in the weak topology ori C( P ) as the conjugate space o L^( T* /^ ) Let A be_ the spectrum o C and let <f be_ a continuous mag of J^ 1 onto A Then there exists a regular measure v on A and an algebraic isomorphism A o C ( P ) onto the algebra o_f all complex- valued bounded measurable functions on ( A , V ) such that: (1) if f e C , then ( A t)(*f ( * )) s f ( 0" ), (2) if f e C ( P ), then /f( ^ )d;u( ^ ) = ^ >A f)( ^ )d'v( c> ). Let f be arbitrary in C( P ) Then there exists a sequence |f j- in DECOMPOSITIONS OP OPERATOR ALGEBRAS. I 59 & which converges weakly to f , so that in particular /f n ( aOg( ^)d/t( JO > ft( *)g( * )djm( eO forge . Defining V aa above it results that ^ n ( tf )gU )d/xU ) = /< Vn> ( ^ 5 (A o g) ( * )d V ( ^ >' where A Q is defined by the equation (A o h)(<p (*)) = h( O, / P for h c . Now Hf n ll is necessarily bounded by a theorem about weakly convergent sequences of linear functionals on Banach spaces, and as by reg- ularity C(A) is dense in L. ( A,v) it follows that {A f } is 1 L o n' weakly convergent in L^fA, V ). As this latter space is weakly sequen- tially complete, there exists a bounded measurable function P on A to which the sequence l^o^n}* * s wea ^7 convergent, and defining A by the equation Af = P, it is clear that for f e & , one of the values of Af is A Q f and that ft ( aOg( * )d/t( S ) = J(t )( c) ) (Ag) ( b )d v ( b ), if f e C (P) and g e . Fixing f, g, and a selection of Ag, it is clear that Af is determined as an element of L^ ( A, v), so that A is single- valued. Thus Af = A f if f e C , so that (1) in the conclu- sion of the lemma is satisfied. It is obvious that for arbitrary f e C(P), /fmd/xU) = /(Af)( cdV( b). It is easily seen that A is linear. To show that A preserves products, let f be in C(P) and g in & , and let f be the weak limit of the sequence {f } in & . Then Af is the weak limit of the sequence {^Af } , and as multiplication is continuous in each factor sepa- rately in the weak topology, we have fg = weak lin^ f g, so A(fg) = weak liin A(f g) and A(f g) =r A(f )A(g) vA(f)A(g), so that n n n n A(fg) = A(f )A(g). By a repetition of the procedure Just utilized, it follows that the last equation is valid for arbitrary f and g in C(P). Now A is univalent, for if Af = 0, then from the fact that for all g 6 , Jt( Og^Jd/uU) = y?Af)( c))(Ag)( 6 )dv(S), it results that Jt( tf)g( / )d^M( ^ ) = for all g (5 . Prom this last equa- tion it is easy to deduce that f = 0. 60 I. E. Segal It remains only to show that A is onto. If F is a bounded measurable function on (A, V ), by regularity there exists a sequence f P n) in C(A) which converges weakly to F. Now if f n = A" 1 ^), from the equation /F n ( & ) (Ag) ( b )d V ( c = Jt n ( if )g( if )dym ( if ) for g e < , it results that the sequence \J*r^ ^ ^( ^ )<*>*( tf ) r has a limit for all g 6T Now ll^ n ll ^ s bounded by the theorem mentioned above, so that ||f If is bounded (for an algebraic isomorphism of a into an algebra of essentially bounded measurable functions pre- serves norms), and it follows readily that the foregoing limit exists for all g e L ( P, yu). Making use again of the weak sequential complete- ness of the conjugate space of an L^-space over a finite measure space, it results that there is an element f in C(P) such that /f n ( aT)g( aOd/aeO ~**Jt( /)g( eOd/A( /) for all g e , . Clearly /(Af)((Ag)( c))dv(c>) = y$(c>)(Ag)(c))dv( b) for all g cdT, from which it is easy to conclude that P = Af . The validity of conclusion (5) of the theorem follows directly from the preceding lemma together with Theorem 1, Example . As an illustration of the use of the preceding result, consider the situation described in Section 8; the invariant measure m is expressed in the form j r m y dju( /), where the m^ are ergodic in- variant measures . Taking Q_ , C 9 and z as in that section, it results that we can also write, for any continuous function f on M f y M f(x)dm(x) = J&tb (Q f )dv( c>), where Q f is the operation of multi- plication by f , which by the same argument as in that section, leads to the equation m(E) = J ^ m 1 ^ (E)dv( c) ) for any Borel set E in M. Here mj is the measure associated with the state T^ , and is ergodio a. a. on ( At V ) f for the inverse image of [6 | m 1 ^ not ergodic J is (V / nif not ergodic J , which has measure zero by Theorem 4 . We note DECOMPOSITIONS OP OPERATOR ALGEBRAS. I 61 also that m^ ^ m 1 ^ if 6 ^ c^ . We conclude by showing that the measure space ( P, JJL ) which occurs in our decompositions can be replaced not only by any equivalent regular compact measure space, as in the preceding ^heorem, but, under a separability restriction, by any equivalent measure space (not necessarily bearing a topology), two measure spaces being regarded as equivalent if there is an algebraic isomorphism between their measure rings. In particu- lar, if separable as a measure space, ( I , JUL ) could be replaced by a measure space over the Borel subsets of the reals, and the decomposition of von Neumann thereby obtained. THEOREM 12. Let & , C, and z be a_s_ in Theorem 1^, and suppose that Q. is. separable (in the uniform topology) and contains the identity operator. Let M = (R, ^ ,r) be a measure space such that C is. alge- braically isomorphic (with preservation o ad joints) t the algebra o all complex-valued bounded measurable functions on M. Then there exists a measure r ! on R, an r-null set R Q , and a map p v u) on R-R t the state space of A such that ; (1) r and r* are absolutely continu- ous with respect t each other; (2) for T e Q. , U> (T) Is a measurable function of p on M; (3) for T e Q and S e C 9 (TSz, z) = cJ (T)U(p)dr f (p), where U(.) is the function on R corresponding to U. Certain parts of the proof of this theorem closely rose iblo the proof of Theorem 1, - we shfili merely sketch these portions. For any fixed T , (STz, z) can in an obvious fashion bo regarded us a continu- ous linear functional on L^lM) (in its norm topolopr<r). Moreover, if S n (.) is a sequence of elements of L^ (M) such that 1 > s n (p) > and S n (p) > Sj^^-j^tp) for all n, and lim n S n (p) = 0, all these conditions holdirg for alnost all p M, then lim P(S ) = 0, where v;e set F = F, To see this, observe that from the given algebraic isomorphism 62 I. E. Segal of C with LQO (M) it results that if S n is the element of C corre- sponding to S n (p), then I > S R > s n +l * It is known that ^ this situation there exists an operator S to which the sequence {. s n j converges strongly, so that Pm(S n ) = ^ S n Tz ' z ^ ' > < STz z )* On the other hand, as an algebraic isomorphism preserves order among the self- ad joint elements, S n (p) > S(p) > a. o., which shows that S(p) =: a. e. and S = 0, so that (STz, z) =0. We next show that for any continuous linear functional <J> on L^M) with the foregoing property, there exists an element f e L. (M) such that <f(k) = y R k(p)f (p)dr(p) for k L^tM). Let s be the func- tion on R defined by the equation s(E) = >(9C,), where % is the characteristic function of E (E R). Now if E = ^i E i' with E^ e 1t and the E 4 mutually disjoint (i = 1,2,...), then % - OC \ ) E. is a 1 E w iin i sequence of functions converging monotone ly to zero. So $(*r, -H n X ) - *0. It follows that s(E) = TT s (E ). Thus s * E *-^i =1 E. i i is count ably- additive j it vanishes on sets of r-measure zero; and it is bounded by || T II ** results from the Radon-Nikodym theorem that s(E) = J^ f(p)dr(p) for some f L n (M). We have f <J>(k) = J^ k(p)f (p)dr(p) when k is a finite linear combination of characteristic functions of sets in l . Now if k is an arbitrary non- negative-valued element of 1^(14), there exists a sequence {l^} of such linear combinations such that k^x) increases monotonely to k a. e., and with k n (x) >. a. e. It follows that the same formula holds for such a k, and hence for all k e L^ (M). Next we show that if f is the element of L^M) defined by the equation (STz, z) = ys(p)f (p)dr(p) , then f (p) > a. e. For if |T I fi(p) = on the set E in f 9 and if S is the element ofc C corre- sponding to the characteristic function of E, then (Sz, z) = clearly, but S 2 = S as (# E ) 2 = X E , so (Sz, z) = //Szj/ 2 and Sz = 0. DECOMPOSITIONS OP OPERATOR ALGEBRAS. I 65 This implies T$ z = for T e Q. or S(Tz) = 0, from which, by the density of dz in H, it follows that S =r 0, so that E must be of measure zero. Now it is easily seen that if <(h) > whenever h is non-negative-valued, then the corresponding function f is a. e. non-nega- tive-valued. Therefore f'(p) > a. e. This shows that r can be replaced in the integration over R by the measure r 1 defined by the equation r (E) =r J^ '(p )dr(p) , E e 1%, . Let f now be defined for T . CL by the equation (STz, z) = /S(p)f T (p)dr ! (p), S6G. If T > 0, (STz, z) is a positive linear functional on C * so f T (p) > a. e. If T and U are arbitrary in Q and if ct is an arbitrary complex number, then it is easily seen that frp^ufp) = f T (p) + ^(p) and tP) = W>t (P ) a. e. We now partially normalize f T (p) by breaking L^ (M) into equivalence classes, two (residue classes of) 'functions 1 (modulo the subspaoe of null functions) being equivalent if they are proportional (relative to constants), then se- lecting one 'function 1 from each equivalence class and then choosing any representative from the corresponding residue class, in an arbitrary fash- ion except for the following restrictions: 1) the representative of a 'function 1 which is proportional to a 'function* which is non-negative a. e. shall be everywhere proportional to a non-negative function; 2) the abso- lute value of the representative at any point shall not exceed the norm of the 'function 1 in L^ (M); 3) the 'functions' which are zero and one a. e. shall have the representatives which are respectively everywhere zero and one; 4) a 'function 1 which is a* e. proportional to a real -valued function shall have a representative which is everywhere proportional to a real-val- ued function. It is clear that a choice of representative can be made sub- ject to these restrictions, and that if g is any representative, assign- ing (Xg as the representative of the 'function' a* e. equal to ctg yields a representative for each element of L^fM). We now assume that f T is a 64 I. E. Segal representative, for all T d . We have now f ocT (p) = Ctf T (p) and * T#T (P) > for a ll P e R and T d . Roughly speaking, it remains only to obtain a null set R o such that for all p R-R Q , and T and U in d , f T<fU (p) ^ f y^) + fjj(p). To do this, let Q Q be a countable SA subring of d , which is dense in Q. , contains I, and which admits multiplication by rationals and by i; such a subring exists because Q. is separable. Then the set of all pairs (U, T), with U and T in Q is countable, and hence there is a null set R Q such that f (p ) = ^ TJ (p) + f T (p) for a11 P e R " R o and all U and T in Q Q . We define tj (T) = f T (p) for T e (2 O and p e R-R O . Now if T is arbitrary in d , we define ^(T) as follows: let {T n } be a sequence in Q Q such that T n * T, and set 6J p (T) - lin^ ^ p (T n ). As |g(p)| < ||g|| for g L^ (M) by virtue of our normalization, l^pfTn - T m )| II T n - T m (| , and clearly V T n-V = f Tn -T m (P) = f T n (P) " f T m (P) = ^p (T n> ' S V' 8 the foregoing limit exists; and it is easily seen to be independent of the sequences used to approximate T, i. e v ^ p (T) is single -valued. It is readily deduced from the additivity of u) on CL Q that for arbitrary T and U in d , ^ (U * T) = cJ (U) o> (T) for p R Q . Now <J (T) is real if T is SA and p ^ R o , for if T n - *> T with the T n in Qu Q , then T' = (1/2) (T + T*) e Q and T 1 > T; and as u) (T' ) is real n n n o n p n for p R , so also is a) (T). Moreover, (J (T#T) > for T e d and p R Q , for if T > T with the T n e Ct Q , then T#T > T*T, and as If OL is real and rational, then plainly u) (ctT) = CtcJ (T) for T e. Q and p R Q . If CX is any real number and sequence of rationals converging to (X , then o^T > (XT so that o) p ( oc n T) - ^ u) ( otT) and hence cu (aT) rr a a) (T) for p ^ J p (iT) r: ia) p (T) for all p and T e Q o , it follows that DECOMPOSITIONS OP OPERATOR ALGEBRAS. I. 65 6c) p (cxT) = acj p (T) for all complex OL and T e Q, Q (p R Q ). It is not difficult to conclude from this by the method just used that the same equation holds for all T 6 Q. . It is obvious that ^^U) ~ 1* and it follows from the fact that u3 p is real on SA operators that cd (T*) = u3 p (T) for any T e Q. . Thus oJ p is a state of Q (p R Q ). To conclude the proof it is sufficient to show that ^ p (T) = f-p(p) a. e. This is true by definition for T Q o . We recall that |/f T || < ||T|| for T Q. . Hence if T C Q, and if ^T n | is a sequence in $ o such that T n > T, then f T ^ f in LOO(M), and so f (p) is a. e. the limit of f (p). As f 'p) = a) (T ) (p i R ) , this shows that x n T n P n r o cJ p (T) = f T (p) a. e. 66 I. E. Segal REFERENCES 1. W. Ambrose, Spectral resolution of groups of unitary operators. Duke Mathematical Journal 11(1944) 589-595. 2. , The I^-system of a unimodular group I. Transactions of the American Mathematical Society 65(1949) 27-48. 2 a . , P. R. Halmos, and S. Kakutanl, The decomposition of meas- ures. Duke Math. Jour. 9(1942) 43-47. 3. J. Dieudonne, Sur le theoreme de Lebesgue-Nikodym III. Annales Univ. Grenoble, Sect. Sci. Math, et Phys. (N. S.) 23(1947-48) 25-53. 4. I. Gelfand and D. A. Raikov, Irreducible unitary representations of locally compact groups. Mat. Sbornik (Rec. Math.) N. S. 13(1943) 301-316 (in Russian). 4a. R. Oodement, Sur la theorie des caracteres. I. Definition et classifi- cation des caracteres. C. R. Acad. Sci. Paris 229(1949) 967-69. 5. P. Mautner, The completeness of the irreducible unitary representa- tions of a locally compact group. Proc. Nat. Acad. Sci. 34(1948) 52-54. 6 , Unitary representations of locally compact groups. Ann. Math. 51(1950) 1-25. 7. ?. J. Murray and J. von Neumann, On rings of operators. Ann. Math. 37 (1936) 116-229. 8. I. E. Segal, Irreducible representations of operator algebras. Bull. Amer. Math. Soc. 53(1947) 73-88. 9. , A class of operator algebras which are determined by groups. Duke Math. Jour. 18(1951) 221-265. 10 , The two-sided regular representation of a unimodular locally compact group. Ann. Math. 51(1950) 293-298. 11. , An extension of Plancherel's formula to separable uni- modular groups. Ann. Math. 52(1950) 272-292. DECOMPOSITIONS OP OPERATOR ALGEBRAS, I. 67 18. and J. von Neumann, A theorem on unitary representations of semisimple Lie groups. Ann. Math. 52(1950) 509-517. 13. M. H. Stone, Linear transformations in Kilbert space. New York 1932, 14. , Application of the theory of Boolean rings to general topology. Trans. Amer. Math. Soc. 41(1937) 375-481. 15. J. von Neumann, On rings of operators. IV. Ann. Math. 41(1940) 94-16L 16. , On rings of operators. Reduction theory. Ann. Math. 50(1949) 401-485. 17. , Zur algebra der funktionaloperationen und Theorie der normalen operatoren. Math. Ann. 102(1930) 370-427. 18. ff9 On some algebraical properties of operator rings. Ann. Math. 44(1943) 709-715. DECOMPOSITIONS OP OPERATOR ALGEBRAS. II: MULTIPLICITY THEORY by I. E. Segal of the University of Chicago 1* Introduction. We determine the most general commutative W* algebra ( ^ weakly closed self-adjoint algebra of bounded linear opera- tors on a Kilbert space) within unitary equivalence. Every such algebra is a direct sum of W-Jfr-algebras of "uniform multiplicity" and an algebra of the latter type of multiplicity n is unltarily equivalent to an n-fold copy (roughly speaking) of a maximal abelian W*-algebra. This last algebra is unitarily equivalent to the algebra of all multiplications by bounded meas- urable functions, of the elements in Lr> over a suitable measure space, and is determined within unitary equivalence by the Boolean ring of measurable subsets modulo the ideal of null sets in the measure apace. Thus to each commutative algebra, there is for each multiplicity (cardinal number) n, a Boolean algebra B(n), and this function B determines the algebra within unitary equivalence; conversely, if B is any such function (vanishing on sufficiently large cardinals), then there exists a commutative Ww-algebra whose multiplicity function is B. The classification by Maharam of Boolean measure rings shows that the measure spaces in question here can be taken to be unions of spaces measure- theoretically identical with the product measure spaces I p , where I is the unit Interval under Lebeague measure and p is a cardinal number (with 1 defined as a one-point space), and allows the replacement of B(n) as a complete unitary invariant by a cardinal-num- ber-valued function ?(p, n) of two arbitrary cardinals giving the number of 2 I. B. Segal copies of IP whose measure ring is a constituent of B(n); and corresponding to any such function there is a commutative W#-algebra. Similar but more limited results are obtained for W-fc-algebras which are not necessarily commutative. As in the commutative case, every Wtf-algebra is a direct sum of W*- algebras of "uniform multiplicity" , and an algebra of the latter type of finite multiplicity n is unitarily equivalent to an n-fold copy of an algebra of uniform multiplicity one. When n is in- finite the last conclusion is invalid except in special cases, notably in that of algebras of "type I". These are algebras which, roughly speaking, are direct integrals of factors of type I, and for them we give a complete structure theory and set of unitary invariants. Specifically, such alge- bras are characterized by the feature that their part of uniform multiplic- ity n is unitarily equivalent to an n-fold copy of a W*--algebra < n of uni- form multiplicity one; and are determined within unitary equivalence by the knowledge for each n of the unitary-equivalence class of the commutative algebra associated with C^ by virtue of the fact that the set of operators commuting with an algebra of uniform multiplicity one is commutative. Most known results in commutative spectral theory either follow readily from the foregoing classification of commutative W#-algebras , or are seen thereby to be equivalent to questions in pure measure theory. Di- rect consequences of our classification (together with the known structure of separable measure spaces) include von Neumann's theorem that on a separ- able Hilbert space, any commutative W^-algebra consists of functions of some operator in the algebra, and the fact that such an algebra is maximal abelian if and only if it has a cyclic vector. Any commutative W#-algebra is algebraically isomorphlc to a maximal abelian W* algebra via a mapping which is weakly blcontinuous and which preserves the operational calculus. The theorem that the W-*- algebra generated by a self-adjoint operator on a separable space consists of all bounded Baire functions of the operator is DECOMPOSITIONS OF OPERATOR ALGEBRAS. II 3 extended to arbitrary spaces, and a brief derivation is given of the Wecken- Plessner-Rokhlin unitary invariants of a self-adjoint operator. Our approach has significant contacts with both the Nakano and Wecken-Plessner-Rokhlin treatments of the multiplicity theory of an indivi- dual operator, as indicated below in the specific instances. In particular, the present definition of algebra of uniform multiplicity for the commuta- tive case is essentially due to Wecken, and the definition which we use in the not necessarily commutative case (while commutative algebras could be treated in terms of this latter definition, which we show to be equivalent to the former definition in the commutative case, it has seemed desirable in v.1ew of the central rfcle of commutative algebras to treat this case sep- arately) is a variation of that of Nakano for abelian rings of projections. Much of the present material (notably Theorems 1-3 and 5-6) was given in a course on spectral theory at the University of Chicago in the Spring term of 1949. We are indebted to members of the course and especial- ly to L. Nachbin for valuable criticisms and suggestions. PART II. COMMUTATIVE ALGEBRAS 2. Definitions and technical preliminaries Definitions 2.1. A W-algebra is a v/eakly closed self -adjoint algebra of (bounded linear) operators (on a Hilbert space). Thruout this paper "operator" will mean "bounded linear operator on a Hilbert space", "Hilbert space" is complex and of arbitrary dimension, and I denotes the identity operator on a Hilbert space which will be clear from the context. An algebra of operators It on a Hilbert space ^ is called an n-fold copy of an algebra G of operators on a Hilbert space rC , n being a car- dinal number greater than 0, if (1) there is a set S of cardinal number n such that fy consists of all functions f on S to K for which the series 2^ x e s ||f(x) || 2 is convergent, with (f, g) defined as 4 I. E. Segal ^ ^ (f(x), g(x)), and (2) d> consists of all operators A of the form X t, S (Af)(x) = Bf(x), for some B in 'Q . (We make the usual convention about infinite sums of complex numbers: they exist only if all but a denumerable number of terms vanish, and if the sum of this denumerable collection exists in the sense of being absolutely convergent). A mas a algebra of operators is one which is maximal abelian in th* algebra of all operators and self-adjoint (i. e. closed under the operation of adjunction). A commutative W-*-algebra (L is said to have uniform multiplicity n, where n is a cardinal number > 0, if it in unitarily equivalent to an n- fold copy of a masa algebra; the algebra consisting of the zero operator only is said to be of uniform multiplicity zero, Definitions 2.2. A measure space is the system composed of a set R, a ring H of subsets of R, and a real non-negative-valued function r on H, such that if C E j3 is a sequence of mutually disjoint elements of H for which the series ^L^rfE^) is convergent, then lj i E. H and r(U^ E^)^ 2_* r(E ), and with the further property that r van- ishes on the void set. If M = (R, "/ , r) is a measure space, a subset V/ of R is called measurable if W E e #, whenever E e H , and W is said to be equivalent to zero if W E is a null set for all E e #, . A measure space is localizable if the lattice of all measurable sets modulo the ideal of sets equivalent to zero is complete. A function on R to a topological space is called measurable if the inverse image of every open set is a measurable set, and two functions are called equivalent if they are equal except on a set equivalent to zero. The Banach space of all com- plex-valued octh-power integrable (complex-valued) functions on M (mod- ulo the subspace of functions equivalent to zero), with the usual norm, is denoted by L^ (M) (1 < ot < oo ); L^ (M) is the space of bounded meas- urable functions, the norm of a function being defined as its essential least upper bound. The Banach algebra whose space is LQQ (M) and in which DECOMPOSITIONS OF OPERATOR ALGEBRAS. II. 5 multiplication is defined in the usual ?/ay is denoted as B(M). The alge- bra of all operations on I^CM) (which denotes the usual Hilbert space, as well as its Banach space) which consist of multiplication by an element of B(M) is denoted by >R(M) and called the multiplication algebra of M. The central results of the part of this paper which deals with commutative algebras can now be stated. THEOREM 1. A maximal abelian self -ad joint algebra of operators on a Hilbert space ig_ unitarily equivalent to_ the algebra of all multipli- cations by bounded measurable functions on the Hilbert space of complex- valued square -intertable functions over an appropriate localizable measure space . \Ve show in a paper ClOj on measure theory to be published separately from the present paper that two masa algebras are unitarily equivalent if and only if they are algebraically isomorphic (in an adjoint- preserving fashion) which in turn is true if and only if the measure rings of the corresponding measure spaces are algebraically isomorphic. By virtue of the Maharam classification of measure rings, the last is the case if and only if the measure spaces have the same cardinal number invariants naturally induced by that classification. Conversely, the multiplication algebra of a localizable measure space is masa (in fact is masa only if the space is localizable). We mention finally that a direct sum of finite meas- ure spaces (see below) is always localizable. THEOREM 2. For any commutative W-*-algebra (L and each cardinal number n > there exists a projection P n in CL such that the least upper bound of the P n (in the lattice o projections) jLs_ the identity operator, and with the contraction of Q, t the range of P n of uniform multiplicity n. There jLs unique such function on the cardinals t the projections in & and the P n are (necessarily) mutually orthogonal . 6 I. E. Segal Before turning to the proof of these theorems we make some fur- ther definitions and remarks* Definition 2.5. A measure space (R, "ft, r) is called (strict- ly) finite (in the present paper)* if R H . It is said to be a regular (locally compact) space if R is a locally compact Hausdorff space, "ft, is contained in the cr-ring generated by the compact subsets of R and con- tains all compact subsets, and if for any E e "/O , r(E) = G.L.B..-j,r(W) = L.U.B.g r(C), where W and C range respectively over the open and the compact subsets of R, which are also in # . For any compact space i 9 C(T*1 denotes the Banach spaces of complex-valued continuous functions on I , with the usual norm. A finite measure space M = (R, ~f?, r) is called perfect if it is regular compact and if for every element of B(M) there is a unique equivalent element of C(R). The system constituted of a complete Boolean ring and a non-negative-valued count ably-additive function on the ring is called a complete measure ring if every element of the ring is the least upper bound of elements of the ring on which the function is finite. If MX - (Rx 9 $ x 9 i\ ) are measure spaces depending on an index X. , and if the R x are mutually disjoint (as can be assumed with- out essential loss of generality), then the direct sum of the M x (over the index set) is the space (R, il , r), where R = U x R x , ~R, is the set of all subsets E of R such that (a) E meets only (at most) count- ably many of the R x , (b) ER^ #^ for all X , (c) H x r (E n R x ) < 005 and for any such set E, r(E) Zl x r(E n R x ) For a discussion of many of the foregoing and related concepts, we refer to [jLOj Definitions 2.4 The spectrum of a commutative complex Banach algebra CL is the topological space consisting of the set of all contin- uous homomorphisms of CL onto the complex numbers, topologized as a rela- tive space of the conjugate space of CU , in its weak topology. An DECOMPOSITIONS OP OPERATOR ALGEBRAS. II 7 "algebraic isomorphism" between two W#-algebras is a correspondence which is an algebraic isomorphism in the usual sense, and with the further prop- erty that if two operators correspond, then so do their adjoints. (In other words, we take the "algebra" of a W-fc-algebra to be an algebra with a distinguished involutory antiautomorphism, viz. that of adjunction). An element 2 of a Hilbert space ~H* is called a cyclic vector for a set x5 f operators on ^ if the set of all Sz, with S e ^J , spans ty (i. e. 7*/ is the smallest closed linear manifold containing those elements). If T?i is a closed linear subspace of fa which is in- variant under >O z is called a relative cyclic vector when the Sz, S e 9 span 7fL . The contraction of ^ to ~Yyt , denoted as A M , is the 1 mt^fi^^mmm . Jft collection of all operators on 7^ of the form x ^ Sx, with S , x) (x e Jft). The set of all operators on ^/ which commute with each element of ^ is called the commutor of xJ and denoted ^ ' . If T is a SA (self -ad joint) operator on a Hilbert space //- with corresponding resolution of the identity {?*.} ( so tna * T ^XdE^ ), the spectral measure associated with T is the function E(.) on the Borel subsets of the reals to the projections on 1^ determined by the condition that it be countably-additive, regular, and that if B is the closed interval (-00 , \] , then E(B) =: E x . If P^ is any family of projections on /*=/ indexed by yU , then (Jju Pyx denotes the least upper bound and I I u. P^ the greatest lower bound of the P^ in the lattice of all projections on ^ . 3. Structure o maximal abelian W^-algebras . We prove Theorem 1 in this section and obtain some incidental results which may be noteworthy. In particular, it is clear from the proof that the measure space in ques- tion in that theorem can be taken to be a direct sum of finite perfect spaces. 8 I. E. Segal LEMMA 1.1. If CL is. a SA algebra of operators on a Hllbert space "fo^ 9 then j^ is. discrete direct sum o sub spaces each o_f which is_ invariant under #/ and has a relative cyclic vector for (L It is clear from Zorn ! s formulation of transfinite induction that there exists a collection L, of mutually orthogonal closed linear sub- spaces of ^ , each of which is invariant under CL and has a relative cyclic vector for Q, 9 and which is maximal with respect to these proper- ties. If ~/6 is the discrete direct sum of the elements of L, , then -fi - ffl 9 for if x is a nonzero element of the orthogonal complement of 7C in /V' * the closure of #x is easily seen to be invariant under & 9 to have the relative cyclic vector x, and to be orthogonal to all the elements of JH, . LEMMA 1.2. A^ commutative W*algebra with & cyclic vector :Ls uni- barily equivalent to_ the algebra of all multiplications b bounded measur- able functions on Lg over a finite perfect measure space . Let Q be a W#-algebra on /*/- with cyclic vector z. It follows from any of a number of representation theorems (see e. g, l\] , Ch. 1) that d is isomorphic as a Banach algebra to (3(P)> where |^ Is the spectrum of Q . For T d , let T(.) denote the corresponding Function on I . Let a) be the functional on Gi defined by the equa- tion CO (T) = (Tz, z), T e Ci 9 and let cO f be the naturally induced Functional on (J(P), so that o) ! (T(.)) - a) (T). Then by the Riesz- flarkoff theorem there is a unique regular measure ^ on I such that cO ! (T(J) = j T( /)d/A(7r), T Q, . It follows from [9] , Theorem 1, bhat (\ U) is a perfect measure space. Now let U be the function on Qz to Lgd^, ju ) defined as Follows: U (Tz) = T(.), - T(.) is a continuous function on the finite DECOMPOSITIONS OP OPERATOR ALGEBRAS. II 9 regular compact measure space (P, >u), and hence square- in tegrable. Now TJ preserves norms, for ||Tz|| 2 = (Tz, Tz) = (T*Tz, z) = u) (T#T) -^ (T*T)( 7Od/u( /) = ^ | T( /) | 2 dyu ( /) . This shows that U Q is single -valued, for if Sz = Tz with S and T in Q 9 then (S-T)z =0, and so (S-T)(.) in Lg(P,yU), 1. e. S(/) - T(/) = a. e., or S(/) = T( /) a. e. As ft z is dense in >+ , and by regularity C(P) is (subject to an obvious identification) dense in L 2 (P, u ), IT O can be uniquely extended to a unitary transformation U on I* to We show next that U takes CL onto t(P, u ), the multiplica- tion algebra of (| , ijj] . This means that the map T ^ UTU""^ on CL will be shown to be onto 7?( (P, LJL). We show actually that UTIT 1 = M T , where Mrp is the operation of multiplication of elements of Lgd , U ) by T(,). The onto character of the map is then an immediate consequence of the fact that (P , JJL ) is perfect. Now as UTU" 1 and M T are both bounded operators on Lg(P> M), it suffices to show that UTTT* 1 * = M^ for x ranging over a dense subset of Lg(P , u ) , e. g. , over 0(Y ) Now if x C(T )* then x = S(.) for some S e CL and IT^x Sz, so UTU -1 x = (TS)(.). Obviously M^oc = T(. )S(. ), which equals (TS)(.) as the map T ^ T(.) is an algebraic isomorphism. COROLLARY 1.1. A commutative W -algebra which has cyclic vector Is maximal abelian. For the multiplication algebra of a finite measure space is maximal abelian (see e.g. D-Q] ). PROOF OF THEOREM. Let G be a masa algebra on /^ f and let >f be the direct sum of the closed linear subspaces /^ , ^ e 1* 9 each of which is invariant under Q and has a cyclic vector for Q 9 - these 10 I. E. Segal exist by Lemma !! The contraction of Q to 7Vv is "by Lemma 1.2 uni- tarily equivalent, say via the unitary transformation U* , to the multi- plication algebra of the finite perfect measure space M^ = ( | * /X ), C T^ n I f = for j f 1 . Now let M = (R, 7? > r) be the direct sum of the Mr- , 6 d,. We define a unitary transformation U on /^ to L 2 (M) as follows: if x c ty, let x = 2Lr e ^ Xf with x,. ^ Hp , and set Ux __ V-? 0/^5 i s ~2i_>c e U x f (this sum exists in the sense of unconditional convergence V 7 1 1 M for the TL- x^ are mutually orthogonal and 2-*^ 1 1 U f x f 1 1 2 1 lx^ || 2 = ||x|| 2 ). It is easily verified that U is, actually, unitary. It remains only to show that the map on the operators in GL to those on I^fM) takes U onto Jft. (M), i. e. that ^t(M) = UtfU* 1 . Let T be arbitrary in Qi. , let T^ be the contraction of T to 1r^ , and let t be the function on R defined as follows: if p efT > then t(p) = T^ (p) (where T^ (.) is the bounded measurable function on Ic corresponding to T, i. e. , Uc T^ Ur^ is the operation of multiplication by IV ( . ) ) . It is easily seen that t(.) is measurable on M, and | I t(. )| | < L.U.B. c c ~ I I Tf (. )| I . Now the bound of the operation of 00 J| c-^X . 3 multiplication by a bounded measurable function on a measure space, of elements of Lg over the space, is readily seen to be identical with the bound of the function in L^ Hence 1 1 T ff (.)( | <||lJr 0* U-l| | = ||r f || Hj 5 1^3 3 _ T|| , so that t is bounded. To show that UTU" 1 is the operation Qt of multiplication by t it is enough, in view of the boundedness of these operators, to show that they agree on a dense subset of Lg(M). In parti- cular it suffices to show that UTtT^f = Q.f if f vanishes outside of a finite union of the 1^ and coincides on each with a continuous function. The foregoing equation is linear in f , and so it is sufficient even to show that UTU -1 f = Qt f for f vanishing outside of I? and DECOMPOSITIONS OF OPERATOR ALGEBRAS. II 11 continuous on 17 ( ^ ). But for such an f, (Q t f )(x) - t (x)f(x), where t (x) = T (x) for x e \l and t* (x) = otherwise; while (UTU-lf)U) = (UTUrl f)(x) (as U-l coincides with U|l on U>k ) = (T^ TUg 1 f)(x) (for UJ 1 f e 7^ , T leaves 7^ invariant, and U agrees with U| on 74g ), = T, (x)f(x) for x ej? and vanishes for x (7 (by the definition of U^ ), and so finally is equal to tl (x)f (x) also. At this point we use, for the first time in the proof, the as- sumption that CL is masa. It follows that Uau*" 1 is likewise masa. As Uau* 1 has been shown to be contained in 7*1 (M ), and as the latter algebra is obviously abelian and SA, it results that UO.TT 1 ZD #f(M), and hence that U4IT 1 = 77? (M). COROLLARY 1.2. A maximal abelian self-adjoint algebra of opera- tors on a_ separable Hilbert space has a cyclic vector. By the known classification of separable measure spaces, every such space has its measure ring isomorphic to that of a finite measure space. Now it is clear that if Lg over a measure space is separable, then so is the measure space, and hence a masa algebra 1*L on a separable Hilbert space 1& is unitarily equivalent to the multiplication algebra of a finite measure space. It is clear that the function identically unity is a cyclic vector for the multiplication algebra of a finite measure space, and hence the corresponding vector in & is cyclic for }#. We remark that the present corollary follows also without the use of the classification theorem for separable measure spaces, from the observation that any collection of mutually orthogonal projections on a separable Hilbert space is at most countable, together with Lemma 2.5 and the remark in the proof of Lemma 2.9 to the effect that a separating vector for a masa algebra is cyclic (the proofs both of the lemma and the remark 12 I. E. Segal being Independent of the corollary). Remark 5,1. The result established in this section essentially includes the spectral theorem, and can be used as a basis for the operation- al calculus. The fact that a commutative W^-algebra is algebraically iso- morphic to a CT( | ) is nearly equivalent to the spectral theorem, and the theorem follows readily in full from Theorem 1. We illustrate the situa- tion by considering the case of a SA operator T on a Hilbert space /^ , the same procedure being valid for any finite number of commuting SA oper- ators. By transfinite induction there exists a masa algebra IK, contain- ing T, and it follows from Theorem 1 that ~frt is unitarily equivalent to the multiplication algebra of a localizable measure space M = (R, 7?, , r). Let T correspond to the operation T f of multiplication by the real- valued bounded measurable function k, let S^ = Lp R I k(p) <. \j , let 3^ be the operation of multiplication by the characteristic function of S A , and let E^ be the operator in M corresponding to Ej^ . It is a straightforward application of known measure theory to verify that T 1 ^/AdEJ in the usual sense (that (T'f, g) - y'* 00 A d (E^ f, g) for any .f and g in I^fM)), and that strong lim c ^ E^ + ^^ E A . It follows that T rr yXdR^ , where {^xj is a resolution of the iden- tity in the usual sense, and it is readily seen that E^ commutes with all operators which commute with T, and that the family C E x) is unique . Now let ~^s be any bounded Baire function on the reals. We shall ahow that )^(T) can be defined simply as the operator in TT correspond- ing to the operation of multiplication by ~X"(k(p)). Again measure theory of a standard sort implies that this operation is </ y( A )dE^ , and it follows that the corresponding operator in 7?L ia < /)^( AJdE^ , and so in particular is independent of the masa algebra in which T is imbedded. DECOMPOSITIONS OP OPERATOR ALGEBRAS. II 13 4. Structure oJF commutative W-fr-algebras We base the proof of Theorem 2 on a series of lemmas. LEMMA 2.1. Let GL be_ a commutative W-fr- algebra of uniform multiplicity n, on Hilbert space 1<f . If P is a nonzero projection in d, then the contraction of d to the range of P likewise has uniform multiplicity n. In proving this, there is evidently no loss of generality in tak- ing CL to be an n-fold copy of the mas a algebra 7ft on Ai (rather than unitarily equivalent to such a copy), so that 7*^ consists of all functions f on a set S, of cardinal number n, to 7* , for which the sum Z> cq l!f(x)|| 2 is convergent, and (f, g) - 21 (f(x), g(x)) for f X t o x G 5 and g in i*t~ ; and d consists of all operators of the form (Af)(x) = Bf(x), with Be>^ . For Be > > let p(B) be the operator in CL defined by the preceding equation. It is clear that <p is an algebraic isomorphism of ^ onto Q, , and hence there is a projection P in >2C such that ^ f>) = P. Putting f for the range of P, ft for the contraction of 7? to 7^ , and # for the contraction of 4L to the range of P, it is not difficult to verify that Q is an n-fold copy of ^f^ onft^. It follows that it is sufficient to prove the lemma for the case n 1. Now assuming n = 1, it must be shown that if W is in the commutor of Q^ 9 then W e d^ (it is clear that Q is SA). Let W be the operator on '^ defined by the equation Wx = WPx, x e 7*^ Then W e d* , for if T and x are arbitrary in Q and 7V" respec- tively, V/Tx = WPTx = WPTPx (for (L is abelian) = WTPx (where T is the contraction of T to P>f) =: TWPx (since W e (Q.) 1 ) = TPWPx (as WPx e PJ*) = TPWPx (by the definition of T) TWPx (since vfrx e, P7*^) = TWx. Hence W e CL , and it follows that the 14 I. E. Segal nj contraction of W to P7^ is in d. 9 and this contraction is W. LEMMA 2.2. Let 7??^ ^5. , mas a algebra on the Hilbert space ^/^ 9 where uu ranges over an index set, and let "^ be_ the direct sum pJF the "ftV Let ^ be_ the set o all operators T on ~fa determined bj such an equation as_ Tx ^ ^y^T^ PM *> where x e A^ , Pyu^ is^ the projection of "^ onto ~fy , Tyu. c ^/^ , and with [ I TyJ I bounded. Then 7# iLs a masa algebra. In the statement of this lemma, and elsewhere when convenient, we make the obvious identification of a summand in a direct sum of Hilbert spaces with a closed linear manifold in the sum. It is readily verified that the algebra ~frt defined in the lemma is SA. To prove the lemma we need therefore only show that if W e 7f(? , then W e 7^ . Prom the defi- nition of 7?f it is clear that P^ e ^ , so that P^ and W commute, which implies that W leaves invariant the subspace IV >^ ~ "/"Cu, Now the operation of contracting an operator to an invariant closed linear manifold is a homomorphism of the algebra of all operators leaving the mani- fold invariant, into the algebra of operators on the manifold. Hence the contraction \^u of W to /l^u commutes with the contraction to the same manifold of each element of ty , and so commutes with every operator in ^/H. As Wu. is maximal abelian, it results that Wyu 6 ^K^ . Now II *>JI ~ N p /< w ll ^ ||w|| , so that llw^.11 is bounded, and it follows that W e 7^ . LEMMA 2.3. If Nm-" iSL & family of mutually orthogonal pro- jections in the commutative W*-algebra ft 9 and If the contraction of Q^ t P^ 7<^ has uniform multiplicity n for all /J. , then the contraction ^o & t ( U/u P^x ) 7V likewise has uniform multiplicity n. Let the contraction ^2^ of Q to P^ /^ be unitarily DECOMPOSITIONS OF OPERATOR ALGEBRAS. II 15 equivalent, via the unitary transformation ILx on V^ty, to an n-fold copy, say C^ on c^u. , of the masa algebra JHu OI * ^C^. . If S^ is the set which here plays the role of the set S in Definition 2.1, all the Syu have cardinal number n, and it is clearly no essential loss of generality to take all the Su, to be identical and equal, say, to S. Let ^ be the direct sum of the ifrL , and let Ttf be the set of all operators T on ^i of the form Tx iLi^T* R^x, x c ~fa , where R^. is the projection of ~^C onto ~fc , T^ C 7^. , and such that flT^I) is bounded. By the preceding lemma 7% masa, and the remainder of the proof consists in showing that 0. Q is unitarily equivalent to an n-fold copy of Tflf on ~K We first define a unitary transformation U on ^p (U/**y )/V" to the n-fold direct sum c of / with itself. For any x C ~hh , U^ Py^ x is some element of <*w , say f^. (.), and IIP^x ||2 x |2 = Hr (a)J|2. Now ||x|| 2 = P/U x 2 , so that the series ^6-u(Z n>u ( a )|| 2 ) * s convergent, and it follows that the series 2^ Q /^ H^/ 1 ^ a ^ll 2 is also conver 8 cnt * Now ?M (* ^ u and the u. * mutually orthogonal subspaces of ^ , so that the Zli n ^T 7 _______ _____ ._ ^. ||fyLc(a)|| ^ implies the convergence of Z-vu. f^ (a) to an element f (a) of # , where Iff (ft))/ 2 = X M Wfyu U)/l 2 - As V ~/ f " " " /" " ' " 2l a e s ||f(a)|| 2 = Z^^-y llf^ (a)|| 2, which last expression is a con- vergent series, we have f e . We define U by the equation Ux = f and observe that U is linear and isometric. To show that U is unitary it remains only to show that it is onto < . Now let g be arbitrary in ^, and let g^ be defined by the equation g^ (a) ~ Ryug(ft) Then 2>a Us/, (a)|| 2 < Z ft ||g(a))| 2 3 o that g/JL ^ , and as Z R^ x = x for all x in ft , we have g = Zuu. . Let yu =- VSt and set y == 2-*j* y^ ; as the y^ are in P^c ^ , they are mutually orthogonal, and the sum ^-yt \\7jm \\ 2 is absolutely convergent as ||^i|| == Il8u| 16 I. E. Segal so that the sum defining y is unconditionally convergent. By the defini- tion of U, (Uy)(a) = ZI^uP^yMa) = 2yi^y^)(a) = ILg^ (a) ]LuR^g(a) = g(a), so Uy = g. Finally we show that UA O U""^ is an n-fold copy of 7^ on ^ *~f We first observe that by the preceding paragraph, c\j is the direct sum of the aL>iji* Now let T be arbitrary in ^2. let T^. be the contraction of T to P*x.A^, and let T^ be the operator U^ I L 1 IT^- on oo^x Then llT^H ll^ll < ||T|| , and as the ^/^ are mutually ortho- gonal, there exists a unique operator T on PC which is an extension of all the T^ . 3y the definition of UL , T^ e QL , and putting ^u. for the map on ^^ to (f^ defined by the equation ( #< (W )f /u )(a) ^ (a), W e'^. , f^ (a) e , we set TJL, = ^^(T^), and put T 1 for the operator on I'C determined by the equation Tx = 21^.1^ R<u x, x e /t^ As c 1 s easily seen to preserve the bounds of operators, ||T M || is bounded and so T f exists and is in We show now that UTU" 1 <fl (T f ), where <p is the map on % to the operators on ^o given by the equation (^ (W ! )f ) (a) Wf(a), f C <^ . As the c^ span < , and as both UTU" 1 and 9^(T f ) are bounded, it suffices to show that UTU-lf^ = ^ (T 1 )^u for all f^ i^, . Now (^(T)f^ )(a) z= T'f^ (a) = TJ< f^ (a). On the other hand, (UTTJ-lf ILA ) (a) (Uu T/x u ^ 1; f/A )(a) = (T^ f^ )(a) =^ (T f^JCa). It remains only to show that every element of ^ has the form OP l(UTU-l) for some T e Q. Q . Let T 1 be an arbitrary element of ^L and let T^, be the contraction of T ! to ^u. ^ et ^u. =: u "^" S^ ( T L. )^/x so that A T^_ is an operator on P^ >/ in A^_ , and let T^ be the (unique) opera- tor on ^r which agrees with T^ on ^ W- , and which annihilates the orthogonal complement of P^. ty- . Now T^x is the contraction of .some ele- ment A of & to P^ W- , and clearly P^ A^u. = T^ , so T^ <S ^2. , As IT'!/ , the directed set DECOMPOSITIONS OP OPERATOR ALGEBRAS. II 17 ^^juep T/x where P ranges over the finite subsets of the >u. f s (or- dered by Inclusion) converges strongly to an operator T on ^ . Obvious- ly T e OL , and It is straightforward to verify that UTU" 1 9*(T'K LEMMA 2.4. If # i. a commutative W*--algebra on #- and if i*L * family o_f projections in Q such that the contraction of d i * V ^ 12. 2 uniform multiplicity n for each tx , then the con- traction of Q t ( U^i P,u ) H- 1.3 likewise o uniform multiplicity n. It is no loss of generality to assume that the index set over which yu. varies is well-ordered. Setting Q,^ P^ (I - U^^ P^ ), then Q ^ C. (J (as the projections in a W--algebra constitute a complete lattice, which is a Boolean algebra when the W-K-algebra is commutative), and as the contraction of u to P^ # is of uniform multiplicity n, it follows from Lemma 2.1 that the contraction of 0. to Q ^ P^ "H- is also of uniform multiplicity n (assigning all multiplicities to an algebra of operators on a zero-dimensional space). It is readily verified that the Q,^ are mutually orthogonal, and hence the preceding lemma implies that the contraction of ft to ( LI/* Qy* )^ is of uniform multiplicity n. Finally, it is not difficult to verify that U^ Q^u. ~ L/^PM The concepts described in the following definition are among those used by Nakano in ?J , and the next lemma is due to Nakano; the proof which v;e give of its non-trivial half is somewhat simpler that that which Nakano Indicates. Definitions 4.1. A W#-algebra Q. on /=/* is called countably- de compos able if every family of mutually orthogonal non-zero projections in Q. is at most countable. An element x in ty is called a separating vector for Q If the only projection P in Q for which Px = is the zero projection; if R is a projection in Q , x is called a relative 18 I. E. Segal separating vector (for Q, on R*f) if x R'H' and x is a separating vector for the contraction of Q to R//'. Remark 4,1. If an element x is a separating vector for CL , then the equation Tx = 0, T e Q. , implies that T 0. For if Tx = 0, then (T*-T)x 0, and it follows that if W is any operator in the W-:: algebra C generated by T#T, then Wx = 0. As x is a separating vector for Q. , v/hich contains C , ^ can contain no nonzero projection. Now any W-x--algebra is generated by the projections it contains, so C - (0), which means T*T = and so finally T = 0. LEMMA 2.5. A commutative W*--algebra is_ countably decomposable if and only if ijt has a separating vector. If Q is a commutative W#-algebra on ~fr with a separating vector x, then Q is countably decomposable, for if Oyu} is any family of mutually orthogonal projections in Q , then ||x || 2 > ^yu'l 1 / 4 x '' 2 so that a ^^- bu ^ at most countably many of the P^ x are zero, which implies that all but at most countably many of the IV are zero, Now let Q. be a countably-decomposable commutative W^-algebra on Jf , and let j* be a family of projections in Q which is maximal with respect to the properties: 1) Q has a relative separating vector on P>/, for P ft 2) the elements of J? are mutually orthogonal. Evi- dently -f is at most countable: let its elements be l?i> * 1*2,...}- and let x^ be a relative separating vector of unit norm for Q. on P^W'. It is easily verified that y = 2_ i 2" i x i is then a relative separating vector of Q on ( (J^P^)~f^. Now Ui p i = I ^ or otherwise there exists a nonzero element z in (I - (J^)1i^ and if QQ is the least upper bound of the projections Q in Q. v/hich are bounded by I - l/iP^ and for which Qz = 0, then d has the relative separating vector z on DECOMPOSITIONS OP OPERATOR ALGEBRAS. II 19 (I - U<P. - Q )>f and I - U P, - Q is orthogonal to all the P, , 110 i 1 o 1 which by the fact that QQ ^ contradicts the maximallty of *& . Hence y is a separating vector for G. on "^ . LEMMA 2,6. If fa is_ a commutative nonzero c ount ably -de compos - able W#-algebra on ty , then there exists a nonzero projection P in Q, such that the contraction of Q t P7<f has uniform multiplicity. Let jj be a set of separating vectors for Q which is maximal with respect to the property that if x and y are any two elements of J , then #x is orthogonal to CLj. Put & x for the closure of &x and P x for the projection of 1^ onto K K . Let (^ be the least upper bound of the projections Q in & for which Q(I - U^P..) 0. and Jv X let be the closed linear subspace of t^ spanned by the ^ x - Then QO ^ 0, for if QO =r 0, then I - U x ? x and the contraction of d to (I - U X P X W *H > is an isomorphism, by the proof of Re- mark 4.1, and putting z for any relative separating vector for CL on JV Q wo (which exists by Lemma 2.5), then z is a separating vector for CL such that Qz is orthogonal to all the Qx with x <^ , - which con- tradicts the maximality of xj . Clearly ^ - U x ^ x , and putting P X = Q Q P X and U x for the range of P X , the contraction Q x of Q. to f^ x has the rela- tive separating vector ^x, and is algebraically isomorphic to the con- traction of d to QQ/^". By Corollary 1.1, # x is masa. Now we utilize a result in [lOJ : if two masa algebras are algebraically isomorphic, then they are unitarily equivalent. It follows that there exists a masa algebra 7^ on a Hilbert space /ti f such that for each x c ^ * x on P X 1^ is unitarily equivalent to ^/ on ~hf, f . Nov/ the direct sum of the PX^' ^ 8 Q^j^f, and it follows without difficulty by a technique previously employed that if n is the cardinal number of d > then the contraction of Q. to 20 I. E. Segal * s unitarily equivalent to an n-f old copy of LEMMA 2.7. I Q is a coinmutative W#- algebra on ty , then there exists a family tX of mutually orthogonal projections in CL whose least upper bound .is I and such that for any P e y , the contraction of fl, t P//- is^ count ably decomposable. Let j^ be a family of projections in # which is maximal with respect to the properties: 1) the elements of y are mutually orthogonal, 2) if P e Jf , then the contraction of Q to PM is countably decompos- able. We show that ^e P ~ I. For otherwise, there exists a non- zero element z in (I-Q)#-, where Q = p ~ P, and putting R Q for the least upper bound of the projections R in CL which are bounded by I-Q and such that Rz 0, I-Q-R Q is orthogonal to all the elements of y , is nonzero as z ^: 0, and Q has the relative separating vector z on (I-Q-R )^, so that by Lemma 2.5 the contraction of & to (I-Q-R O )#- is countably decomposable. LEMMA 2,8. Let Q k *L commutative WK- algebra containing I. For each cardinal number n ^ let B^ be_ the least upper bound of the projections S in Q such that the contraction o A t SH" has uni- form multiplicity n. Then U n R n = I. Set Q ^: U n Rn and as the basis of an indirect proof, assume, Q yd I. Let the contraction of Q. to (I-Q)#" be denoted as ^2^, and let P^ be a nonzero projection in (2^ such that the contraction d^ of $- to P (I-Q)>f is countably-decomposable; by Lemma 2.7, P^ exists, and P^I-QJW- p 0. IT P I is the contraction to (I-Q)#- of the pro- jection P in d (that such a projection P exists is easily verified), then we can write P-^I-Q)/^ = Pd-Q)^. By Lemma 2.6, there exists a nonzero projection Ng in #g such that the contraction of Ob^ to DECOMPOSITIONS OP OPERATOR ALGEBRAS. II 21 N 2 P(I-Q)>f has uniform multiplicity, say n, and obviously NgPd-Q)/^? 6 0. Now if N^ is a projection in <3- whose contraction to P(I-Q)#- is N 2 , and if N is a projection in Q whose contraction to (I-Q)#- is N^ (the existence of these projections being clear), then the contraction of # 2 to N 2 P(I-Q)W- is the same as the contraction of Q to NP(I-Q)#-. It follows that NP(I-Q) < R n or NP(I-Q)R n = NP(I-Q), which by the defi- nitions of R R and Q, implies that NP(I-Q) = 0, a contradiction. LEMMA 2.9. Let # b. . commutative W-x algebra of uniform multi- plicity n and also of uniform multiplicity m. Then nH, = m K . If P is a nonzero projection in the algebra of operators (^ on ^ such that the contraction d^ of Q to P// 1 is countably decom- posable, then by Lemma 2.1, fl^ is of uniform multiplicities both n and m. Hence by Lemma 2.7 it suffices to prove the present lemma under the as- sumption (which we now make) that Q. is countably decomposable. Let ( be unitarily equivalent on the one hand to an n-fold copy of the masa alge- bra T^f on K , and on the other to an m-fold copy of the masa algebra *% on . Now d. is algebraically isomorohic to both 7? and JL , and hence both of these algebras are countably decomposable. By Lemma 2.5 thare exist separating vectors u and v for T^f on / and V on ^ re* spectively. Now the closure of tyu is invariant under >?P so that the projection P on this closure commutes with each element of T^f . As ?7 is maximal abellan, P . 7% . Obviously (I-P)u = 0, and since u is a separating vector for 7/f , I-P 0, which shows that the closure of b(u is 7^ , i. e., u is a cyclic vector for ^. Similarly v is a cyclic vector for >f on ^6 It follov/s from the definition of uniform multiplicity that /^ is the direct sum of n mutually orthogonal subspaces ft ( /A S, 2 I. E. Segal where S Is an index set of cardinal number n) each of which is invariant tinder Ot and such that the contraction of Q to ft^u, is unitarily equivalent to T^t on ^ ; and ~f is also the direct sum of m mutually orthogonal subspaces v ( v e T, T being an index set of cardinal m), where &- leaves each of the v invariant and whose contraction to any oj, is unitarily equivalent to ^ on <> . By the preceding paragraph, there exist vectors x^. >^ and j v e v such that ?XILX is dense in 7C and $y v is dense in & v . Now the projection x^v of Xi>. onto <^ v vanishes, except for countably many "V , so that the set of all indices V for which x^ v ^ for some yu has cardinal number at most ^ n. But for every V there is a yu such that x,, v j 0, for otherwise, taking X to be an index in T such that x^ for all /A , then clearly xu is orthogonal to ^ x for all yu e S. In particular, (x^ , Ay x ) ~ for all A e # and /^ S, or (Attx.^ , y^ ) = for A & and ^u e S. Now the A-fcx^ span % as A ranges over d , and so it follows from the last equation that (z^ , y x ) = for all z^ in A^x . Since the 7^ span "H* , it results that (z, y^ ) = for all z e 7^- , which implies y x 0, a contradiction. Thus the cardinal number m of T is at most K n. By symmetry, n < K m, and it follows that n K = m K . LEMMA 2.10. Let "% be a countably decomposable mas a algebra of operators on the Hilbert space ft . Let x be a cyclic vector for J*t and let {ViJ be an arbitrary sequence o^f vectors in "/^ . Then there exists * nonzero projection P in ty such that Py ?^fx (1=1, 2,...). We note that as shown in the proof of Lemma 2.9, there does ac- tually exist a cyclic vector for 7?f on ^ (in fact^ any separating vector is such). By Lemma 1.2, we may assume that 7^ is the algebra of all multiplications by bounded measurable functions on Lg over a finite DECOMPOSITIONS Op OPERATOR ALGEBRAS. II 23 measure space M = (R, 7? r) It is clear from the fact that Tflfx is dense in Lg(M) that x(a) 7^ a. e. on M. Now setting E. . ~ [a | (^(a)) < J |x(a)| ] , it follows that Uj E differs from R by a set of measure zero, and hence there is a J^ such that r(R - E. ) < r(R)2" 1 " 1 . Putting E ~ f) E , it results that r(R - E) < ZI. r(R - E ) < (l/2)r(R). Now taking P to be the operation of multiplication by the characteristic function of E, we have Py. T^x, for Py^ is the product of x and the function, bounded by J., which equals y^(a) (x(a) )" for a * E and equals zero elsewhere. Our proof of the next lemma utilizes a simplification of a device employed by Nakano C^] in connection with a similar result for the case of separable Hilbert spaces. LEMMA 2.11. Suppose commutative Wft- algebra has uniform multi- plicities both n and m, where n Ijs finite and m < t . Then n = m. As in the proof of Lemma 2.9, we can confine our attention to the case in which Q. is countably decomposable and obtain subspaces ^x and < v of the space 7/* on which O. acts, and vectors x and y y in 7^ and of^ such that Qx.^ and ?y v are dense in ^^ and *o v respectively, and with ^ the direct sum of the Tr^ and also the direct sum of the ^ t (here u = l,2,...,n and v = l,2,...,m if m is / finite and ( ~u = 1,2,... otherwise). Putting V^ for a unitary trans- formation from / onto A^u which implements the equivalence of 7% and the contraction of 0. to ft^ , and x 1 for a cyclic vector for % , we can clearly take x.^ = U^ x 1 without essential loss of generality. Now putting j v/UL for the projection of y v onto ^^ , we evidently have y v Zl^ y v/4 . Now setting y' ir U^^v^t 4t results from the preceding lemma that there exists a nonzero projection P 1 in >%, such that P'yJL ^fx 1 . Now putting P for the projection in GL which is 24 I. E. Segal jmltarily equivalent via the given transformation to the n-fold copy of P 1 (i. e. the contraction of P to ^^ is ^ P'TJ^ 1 ), then P ^= 0. It Ls easily seen from the relation P'yJL e % x ! that Py e Qj^ For all /* and V , say Py v/x T^ JL^ ; multiplying this equation jy P shows that we can suppose T^ = PT^ It follows that Py v = ^v= l T ^M x^ , for all V . Now we issume m > n and derive a contradiction. We use the fact that in an r- limensional module over a commutative ring with unit, any r + 1 elements ire linearly dependent over the ring (see 4] , Th.51). We apply this to the module over P (L of all ordered n-tuples of elements of P Q_ , and in >articular to the n f 1 n-tuples (T, T, ..., T ^ ^ ^ = 1>2 ' * Ct results that there exist elements S. , S , .*., S of P Q- which are J- 2 n+1 lot all zero and such that JJ *+* T v/4 S v = 0. Hence 21 n * 1 S v = or E ^ s v ^v = 0- As s v 7v 6 ci v *n d since the ire mutually orthogonal, we have S v y v = 0. Ilie circumstance that y v Ls a separating vector for Q_ now implies that S v ^ (v=l,..., ), a contradiction. LEMMA 2.12, With the notation of Lemma 2.8, the ^ are mutually >rthogonal. For if RjjB^ T: 0, the contraction of Q to 1^7^ is of uni- "orm multiplicity m by Lemma 2.4, so that by Lemma 2.1, the contraction >f Ct to ^(fW^) likewise has uniform multiplicity m. By symmetry, ;he same contraction also has uniform multiplicity n. It follows from jemma 2.10, that either m n or else one of m and n is finite and ihe other is not greater than < o . By Lemma 2.11, m = n in the latter tase also. PROOF OF THEOREM. With ^ as in Lemma 2.8, we have, putting lo for I-E, where E is the maximal projection in CL 9 vJ n Rn = I by DECOMPOSITIONS OP OPERATOR ALGEBRAS. II 25 Lemma 2.8, and the contraction of d to ^ W* has uniform multiplicity n by Lemma 2.4. Now if P n is for each cardinal n a projection in & with the properties stated in the theorem, then from the definition of R n , it is clear that P n < R^. Now (J P =1, but the R R are mutually disjoint by the preceding lemma, so that P n R = for m ^ n, and it results that R ft = P . 5 Unitary invariants o commutative W-algebraa and o SA opera- tors. In this section we first prove a theorem which gives a simple com- plete set of unitary invariants for a commutative W-fr- algebra. Assuming, in order to avoid a trivial complication, that the identity is in the algebra, these invariants consist of Boolean rings B(n), one such ring being attach- ed to each cardinal number (or multiplicity) n, and vanishing for suffi- ciently large n. These rings are (lattice -theoretically) complete measure rings, and all such rings may occur. The classification theorem of Maharam 3 ] for measure rings is stated for the CT-finite case, but there is no difficulty in extending it to an arbitrary complete measure ring. The use of this extended classification provides a still simpler set of invariants, consisting essentially of a function on pairs of cardinals to the cardinals, - if f is this function, f (m, n) is the number of direct summands of the measure ring of the infinite product measure space I m , where I is the unit interval under Lebesgue measure, which occur in (the direct decomposi- tion into homogeneous parts of) B(n), but the discrete part of B(n) and the case 1 < m < ^ must be treated separately. The validity of these invariants, whose range is clear, follows at once from the following theo- rem together with Maharam f s theorem, and we refer to 3 ], from which the mode of derivation of these bardinals is clear. Thus the most general commutative W-x-algebra can be regarded as completely and rather explicitly known. 26 I. E. Segal Definition 5.1. Let 4. be a commutative W#- algebra, and let (PR! be as in Theorem 2. The Boolean ring B(n) of all projections in the contraction of CL to the range of P n (which ring is shown below to be a complete me a sure -bear ing ring) is called the measure ring of CL for the multiplicity n. THEOREM 5. Two commutative W*-algebras are unitarily equivalent ! and only if their measure rings for the same multiplicities are algebra- ically iaomorphic. and also the maximal (necessarily closed) linear mani- folds which they annihilate have the same dimensions* For any commutative W*algebra CL , the contraction of d to P n # , where & is the space on which & acts and P n is as in Theorem 2, will be called the part of d of uniform multiplicity n. Now if O and are unitarily equivalent W#- algebras it is clear from Theorem 2 that their parts C^ and o# n of uniform multiplicity n are unitarily equivalent, and hence their measure rings for the same multiplicity are algebraically isomorphic. It is obvious that the dimensions of the maximal closed linear manifolds which C an <} *U annihilate are equal. Now suppose that C and & are commutative W#-algebras whose measure rings for the same multiplicities are algebraically isomorphic, and such that the maximal linear manifolds which they annihilate have the same dimension. We shall show that C and <ff are unitarily equivalent, and for this purpose we may evidently assume that both C and & contain the identity operators on the respective spaces on which they act. Let C n and n be the parts of C and of uniform multiplicity n, and let 7^ n and ? n be mas a algebras, to n-fold copies of which C n and & ^ are respectively unitarily equivalent* Then it is clear from the defini- tion of n-fold copy that C n and >^f n on the one hand and f and ?7 on the other, are algebraically isomorphic. Now 77f n and ){. n are unique DECOMPOSITIONS OP OPERATOR ALGEBRAS. II 27 within unitary equivalence, for taking the case of "# n , If C n is also unltarily equivalent to an n-fold copy of the masa algebra < n , then C n and - n are algebraically isomorphic and so ^ n and n are algebrai- cally isomorphic. Now 7^ and 7 are both multiplication algebras of localizable spaces, within unitary equivalence, by Theorem 1, and it is shown in [lo] that if two such algebras are algebraically isomorphic, then they are unitarily equivalent. Let W n and ~% n be respectively (unitarily equivalent to) the multiplication algebras of the localizable measure spaces M^ and N n . The Boolean ring of projections in (3 n is plainly algebraically isomorphic with the ring of projections In 7^ n which in turn is readily seen to be isomorphic with the measure ring of M^ Similarly the Boolean ring of pro- jections in f n is algebraically isomorphic to the measure ring of N n . Hence ^ and ? have algebraically Isomorphic measure rings. By a result in (_loj their multiplication algebras are then unitarily equiva- lent. Thus ^f n and >6 are unitarily equivalent, and it follows that n and Jff are unitarily equivalent. It is straightforward to show from this that C and JL? are unitarily equivalent. Next we obtain a complete set of unitary invariants for a SA oper- ator, this set being due to Wecken [14] and to Plessner and Rokhlin [&] , some of whose techniques we use* Before stating the basic theorem we use the foregoing theory to reduce the problem to the situation treated in the theorem. If T is a SA operator, and if T n is its part of uniform multi- plicity n ( s contraction of T to the range of P n , where P n is as in Theorem 2, d being the W#-algebra generated by T), then T n is uni- tarily equivalent to an n-fold copy of an operator S n with simple spec- trum (i. e. the Wa-algebra generated by S n is masa). By Theorem 1, S n can be taken to be the operation of multiplication by some function, on ), for some localizable measure space U^. It is easily seen that a 28 I. E. Segal complete set of unitary invariants for the S n is also a complete set for T, and so the problem is reduced to the essentially measure -the ore tic one of determining when two multiplication operators, each of which has simple spectrum, are unitarily equivalent. The classification of Maharam could be used to reduce the problem further to the case when the measure spaces in question are homogeneous. Naturally it is a restriction on a measure space for it to admit a multiplication operator with simple spectrum, but we shall not discuss the nature of this restriction, which at present is un- clear (except for the fact, which follows from Corollary 5.3 without diffi- culty, that the separability character of the space must not exceed the cardinality of the continuum). Thus in order to obtain a complete set of unitary invariants for a SA operator, it is sufficient, in view of the foregoing, to obtain such a set for SA operators with simple spectrum, and in the remainder of this section we consider only such operators. We note that if attention is restricted to SA operators with simple spectra which are unitarily equiva- lent to multiplication operators on finite measure spaces (and for opera- tors on separable Hilbert spaces this is always the case, as it means that the W-*-algebra generated by the operator is countably decomposable), the operator is determined within unitary equivalence by its spectrum together with its spectral null sets, - for separable Hilbert spaces this was proved by Nakano [ 6 J The invariants given by the following theorem for the general case are a kind of generalization of these invariants. Another set of invariants for the general case, more closely related to those for the case of finite measure spaces, but in some respects more complicated than the present ones is due to Nakano [7] . Definition 5.2. For an arbitrary SA operator T on a Hilbert space #" , the weifftitad spectrum C(T) is the family of all (finite regu- lar) measures m on the reals of the form m(B) - (E(B)x, x), where B DECOMPOSITIONS OF OPERATOR ALGEBRAS* II 29 is an arbitrary Borel subset of the reals, E(.) is the spectral measure associated with T and x is arbitrary in W . (It is easily seen that m is concentrated on the spectrum of T, as this term is usually defined). THEOREM 4. (Wecken-Plessner-Rokhlin) . Two SA operators on Hilbert spaces with simple spectra are unitarily equivalent i and only If their weighted spectra are the same* It is clear that if two SA operators are unitarily equivalent, then their weighted spectra are the same. Now let T and T 1 be SA oper- ators with simple spectra on Hilbert spaces ty and -ft* respectively whose weighted spectra <S and G f are the same. To show that T and T 1 are unitarily equivalent ?/e require two lemmas, which are essentially contained in the work of the authors mentioned. In connection with these lemmas we recall that two measures (on the same ring of sets) are said to be orthogonal if the only measure absolutely continuous with respect to both of them is the zero measure. LEMMA 4.1. Let x and y be elements of ~ty 9 let #. be_ the W*~ algebra generated by_ T , and let E ( , ) be_ the spectral measure associ- ated with T. Then & x is orthogonal to #y if and only if m x is. orthogonal to m_., where for any z e ty , m z i the measure on the Borel subsets f the reals given bj the equation m z (B) = (E(B)z, z). We observe to begin with that if z is in the closure fa x ^ $ x , then m z is absolutely continuous with respect to m^. For if {V n } is a sequence in d such that V n x > z, then m z (B) = ||E(B)x||2 = linijJlEfB^xllS =: lii^ ||v n E(B)x || 2, and if m^B) = we have E(B)x = and it results that m z (B) = 0. We note also that the projection P x of 7^ onto / x is in Q. for as / x is invariant under d , P x e Q* , but & 1 = Q Similarly the projection P y of 50 I. E. Segal i^ onto the closure 1^* of ^7 is in & . Thus P x and P_ commute. Now suppose that n^ is orthogonal to m-.. By the last observa- tion, to show that ~A? and % are orthogonal it suffices to show that their intersection is 0. Now if z e "fc. n fa , then as shown in the pre- ~ y ceding paragraph m z is orthogonal to both m and m , so m = 0, from which it follows trivially that z = 0. Next we assume that 0.x is orthogonal to ^2y and show that then m^ and nr are orthogonal. As the basis of an indirect proof, let n be a nonzero finite regular measure on the reals which is absolutely continuous with respect to both m^ and my. By the Radon-Nikodym theorem, there exist BL, and m. integrable non -negative functions h_ and h__ r r respectively such that n(B) = J h x ( Xjdm^ x) =r y B h-( X)dniy( /V), where B is an arbitrary Borel set. Putting g x ( X) = min fl, h x (X)} and g^( X) = min l, h.( X.)j , and setting n x and n- for the set ~y y J functions defined by the equations n x (B) == y B g x ( Xjdm^f A) and ny(B) = o/B Syf^-J^yf^)* then it is clear that n and n^ are absolutely con- tinuous with respect to each other, that the same is true of n and n , and that for any Borel set B we have both ^(B) ^ ^(B) and ny(B) niy(B). In particular, n x and n- are absolutely continuous with respect to each other and so by the Radon-Nikodym theorem we have ny(B) = t/B f ( Xjdn^ X), for some n x - integrable function f. Defining m on Borel sets B by the equation m(B) = j^ f(X)dn x (X)f where f f (?O =r min {l, f(A.)| , It is evident that m(B) < ny(B), m(B) < m^(B) 9 and m^ 0. Thus m is a nonzero regular measure on the reals such that m(B) xajjB) and m(B) < my(B) for all B. Applying the Radon-Nikodym theorem once more, we have m(B) y B f x (X)dm x (X) = t/B f y ( X.)dmy( X), where f x and f y are n^ and my integrable functions respectively, which are bounded by unity. As m x and my are regular measures, f x and fy can be taken to be Baire DECOMPOSITIONS OP OPERATOR ALGEBRAS. II 31 functions. Now f x (T) is a positive semidefinite SA operator and so has the form V 2 for some SA operator V In & Clearly m(B) - f^ f x ( X)d(E x x, x) = (f x (T)E(B)x, x) = (V 2 E(B)x, x) = (E(B)Vx, Vx) :z rn^CB) where x 1 Vx. Similarly m = ni , for some y 1 fc Qy Thus (E(B)xS x f ) = (E(B)y', y' ) for all Borel sets B. As the E(B) generate CL in the strong operator topology, it follows that (Sx 1 , x 1 ) (Sy ! , y 1 ) for all operators S in d . Taking S to be first the projection on /t' and next to be the projection on # -. shows that x f x j ~ y f r: 0, so m = 0, a contradiction. LEMMA 4*2. Let x be^ an arbitrary nonzero element of H* . The contraction o T t the closure ojT <2x is_ unitarily equivalent to the operation of multiplication by the coordinate function X on L 2 (M X ), where M x i the regular measure space on the reals * -co < X < oo , with measure in . A proof of this can be given which is a straightforward adapta- tion and simplification of the proof of Lemma 1.2 and we omit further de- tails. Completion of proof of theorem. Let y be a subset of the weighted spectrum 6^(T) of T which is maximal with respect to its ele- ments being mutually orthogonal and nonzero; the existence of J^ is clear from transfinite induction. For each /> ^ , let x (= x(yo)) and X T (= x'(^o)) be elements of ~ty and TV 1 respectively such that ^ (B) = (E(B)x, x) =: (E f (B)xS * f ), where E'(.) is the spectral meas- ure associated with T 1 . We show now that the Qx^o span "ty, X) c ^ Let > be the closed linear manifold spanned by the &x/) 9 fl & 7 > &Il ^ L let d be its orthogonal complement in H" Clearly ^t is invariant under Q , and if z is any nonzero element of S, m z c C^(T) and n^ is orthogonal to all the m x with f> C 3s by Lemma 4.1. Hence >tf = 0. 32 I. E. Segal Thus //* Is the direct sum of the "/^ ( /> e ^), where 7-^ is the closure of (Jxy, , and by symmetry /^' is the direct sum of the 'Hfl ^ fl e 'y)* where //' is the closure of ^i^j, , fl,^ being the W*-algebra generated by T f . By Lemma 4.2, the contraction of T to ^, is unitarily equivalent to the operation of multiplication by the coordi- nate function on L_(M ) and hence equivalent to the contraction of T f <5 X/O to 7^ . It follows without difficulty that T and T 1 are unitarily equivalent. 6. Applications. In this section we make a number of applica- tions of the preceding structure theory, mainly to spectral theory. Our basic result is as follows. THEOREM 5. If CL ifi, L commutative Wtt-algebra there is. a maxi- mal abelian W-algebra 7 tx> which d is^ algebraically isomorphic (with preservation of ad Joints ) . Any such isomorphism <f of (L onto 7^( ll bicontinuous in the weak topology and has the property that lf_ f is any bounded Baire function on the complex numbers , then for any operator T in d 9 J^(f(T))= f(f(T)). The algebra 7^ is unique within unitary equivalence, and the dimension of the space on which l_t acts is_ not greater than the corresponding dimension for d . If J^i and <jP 2 are algebraic isomorphisms of the W-x--algebra Q onto a masa algebra ^ , then ^ = < f-\ ( f > '' l3 *** algebraic isomor- phism of /?f onto itself. As an algebraic isomorphism between masa alge- bras is induced by some unitary transformation between the corresponding Hilbert spaces fioj , there is a unitary operator U on the Hilbert space /^ on which y% acts, such that ]^(T) = U*TU, T e ^. Hence to show that any isomorphism of Q onto )^ is bicontinuous in the weak .topology, and preserves the operational calculus, it suffices to show that any one DECOMPOSITIONS OP OPERATOR ALGEBRAS. II 33 isomorphism has these properties. Moreover, if CL is algebraically iso- morphic to a masa algebra > , by the result just quoted >^ and 7 are unitarily equivalent, i. e v % is essentially unique. Hence to conclude the proof of the theorem it suffices to show that for the given W-fr-algebra d , there exists an isomorphism of #. onto a masa algebra P*( which is weakly bicontinuous and preserves the operational calculus. Let P n be as in Theorem 2, and let the contraction of CL to P n /*/- (n > 0) be unitarily equivalent to an n-fold copy of the raasa alge- bra ^ on 1-^ . By Theorem 1 we may assume that 1^_ ' consists of L " n n 2 over a localizable measure space M^ =: (R , 72 , r ) and that ^f is n n n n n the multiplication algebra of M^ Let the measure space M = (R, 1%, , r) be the direct sum of the M , for all n, so that L (M) can be identified n 2 in a clear fashion with the direct sum /t^ of the ^ n , and let Tff be the algebra on H corresponding to the multiplication algebra of M. Then ^ is readily seen to consist of all operators T whose contraction T n to ^n is in ^- and such that ^ T ^ is bounded - Now ^ is masa, for if T e 7^ f , and if Q is the projection of / onto /^ n * then as plainly Q^ e >( , T^ = Q T so that T leaves ~fr invariant. Putting T for the contraction of T to 7 * then (UV) = U V for n n n n n any operators U and V leaving /^ n invariant, and it follows that T n S n = S R T for S e ^. Hence T n e )^i , and as obviously |JTj| < ||T|| , we have T e 7^. (The masa character of 7/( also follows directly from a result in LlOj ) Next we define an isomorphism <p ot Q. onto >^. For any operator T in Q_ , let T denote its contraction to P >fv and for any operator U in the contraction of & to P n >f* let ^(U) *>e the oper- ator in 7^ n whose n-fold copy U is taken into by the unitary equivalence of d with the n-fold copy of ty n . It is easily seen that 11^(^)11 = llull , so that there exists a (clearly unique) operator ^ ( T ) on ^ 34 I. E. Segal whose contraction to ^ n is X^ n (T n ), and evidently <f (T) e ^. Now "Y^ n is an isomorphism, and it follows without difficulty that so also is f To see that 9^ * s we & k ly continuous it is sufficient, by virtue of the linearity of <]P , to show that it is weakly continuous at 0. Now [T c a | ICr^v 7 i } ' < 6 ' i e P J ' where c >0 ' p ls finlte > and the x^ and y. are elements of ^ , is an arbitrary neighborhood of in d in the weak topology. A general neighborhood of in ^K, in the weak topology is ^T' c)^ J (T'x, yj ) j < , i e p] , where and P are as before and the xj and y are in % . For any x in # n we define yQjjfx) to be the element of P #" obtained as follows: let P n >/" be the direct sum of the n closed linear subspaces // (j S , where S is an index set of cardinal n), where d leaves 1*^* in- variant and has a contraction on /y- unltarily equivalent, via the uni- tary operator U n .i on ^ to //- , 9 to Ttf on ^ (theae subspaces exist because the contraction of d to P n "^ has uniform multiplicity n}. Let J n be any element of S n , and set jo (x 1 ) = U n x f , x f e 7t^ n . Now for any element x e T we put yO (x) = 2I n /^(^ ) where x is the projection of x onto ^ n * It is easily seen that for any x 1 and y' in /ti n and OM c 7# n , (T'x', y' ) = ( ^^(T 1 ) i /7 n (x l ), /? n (y f )), and it follows that for any x' and y in # and T in )^ , ^^(x 1 ), /o (y f ))> where x 1 and y 1 and the projections of NT" n J n n n n x 1 and y ! on ^/ and T 1 is the contraction of T f to ^ . Summing both sides of the last equation over n shows that (T'x 1 , y 1 ) = (^^(T'JyOfx' ), yofy 1 ))* Hence the inverse image under ^ of (T'c ^ | |(T'X, y')/< e , i P] is [T ft| Mv^iV^i^ < ,1 FJ , and so is a neighborhood of in d . Next we show that ^ -1 is weakly continuous. It is not diffi- cult to see from the definition of uniform multiplicity that the contraction DECOMPOSITIONS OP OPERATOR ALGEBRAS. II 36 of d to P n # is unitarily equivalent, say via the unitary transforma- tion V n on P #", to the algebra of all multiplications by bounded meas- urable functions on M , on L (M X W ), where W is the measure space " n n n whose set is S n and in which each finite subset is measurable and has measure equal to its cardinal. Now if x and y are arbitrary in P 7V", say V n x = x(p, i) and V n y =: y(p, i) (p e R^ i , S n ) and T is arbitrary in (2 , then v n T n V~ is the operation of multiplication by a bounded measurable function ^(p), where T n is the contraction of T to P n ^, and (Tx, y) = f. t (p)x(p, i)y(p, i)dr(p)di, and integrating c/ 1^ X W n n first with respect to i, this equals /. t (p)w(p )dr(p) 9 where w(p) = r yM n n 1 S X * P| ^yte* *)' so w 1^(1^). Writing w in the form w(p) = x(p)y ! (p), with x f and y in Lg^) and with |)x ! || 2 = |\y'|| 2 = llwll x , it results that (Tx, y) = (Tx f , y f )> where T' = Now for arbitrary x and y in 1^ we write x = 2 n x n and y = ^n v n* w ^^ n ^ and y in P ^, and summing both sides of the equa- tion (T^, y n ) = (Ttx^, y^) over n, then (Tx, y) = (T'x, y'), where x 1 n . x 1 and y f = n ^ f * ^ nese sums existing in the sense of unconditional convergence because -2^ n Jl^l' 2 I= /^ , i)y n (p, dr(p) = nx W n (Z n IU n |j 2) ( 2I n ||y n || 2 ) = |U1| ||y|) , which is finite; and similarly when x is replaced by y. Hence the image under ^ of the neighborhood [T e d\ |(Tx i , 7 )\< , i p] of in d is of the form [T* C >/| |(Tx , yt )J < CJ , with the x and the yj[ in K, and so is a neighborhood of in 7% . Thus ^>-l is weakly continuous. It remains only to show that <f> (f(T)) = f ( <f(T)) for any bounded Baire function f , and this we show is valid for any weakly contin- uous algebraic homomorphism <jP . The foregoing equation is obviously valid when f is a polynomial, and it follows from the We iers trass approximation 36 I. E. Segal theorem that it is then valid for any continuous function f (in view of the boundedness of the spectra of T and <f (T)). Now let Jf be the collection of all bounded Baire functions for which that equation is valid (for all T (2.): we show that ^ is closed under bounded pointwise convergence of sequences. Let ( f } be a sequence in *-f such that f n ( <* ) * f ( * ) for all complex <x , and with l f n (- ) I bounded (n = 1,2,...). It follows from the spectral theorem for normal operators to- gether with the Lebesgue convergence theorem that the sequence l^ n ( T )} converges weakly to f(T), and similarly f (SP( T ))} converges weakly to f(f(T)). As <p is weakly continuous it results that <p(f(T)) f(<P(T)), and hence ^ contains all bounded Baire functions, for it is clear that all such functions are in the smallest collection of functions containing all bounded continuous functions and closed under bounded point- wise convergence. The following result is due originally to von Neumann [is] COROLLARY 5.1. For any commutative W-fr-algebra d on a separable Hilbert space there is_ a self-adjoint element of Q such that every ele- ment of d is. . Baire function of T. Let 7^ be a masa algebra on 7t^ to which d is algebraically isomorphic. By Theorem 1, ^ is unitarily equivalent to the multiplication algebra of some localizable measure space M, and by the preceding theorem Lg(M), which can be identified with % , is separable. By the known classi- fication of separable measure spaces, M can be taken to be (i. e. has its measure ring isomorphic to that of) the direct sum of a (possibly vacuous) real bounded interval under Lebesgue measure and a (possibly vacuous) dis- crete measure space containing at most a countable number of paints, which can be assumed to lie in some real bounded interval disjoint from the previ- ous one, and to have finite total measure. The resulting measure space is DECOMPOSITIONS OP OPERATOR ALGEBRAS. II 37 finite and regular, and for every measurable function on such a space there Is a Baire function equal a. e. to it. It follows that every multiplica- tion on Lg(M) by a bounded measurable function is a bounded Baire func- tion of the operation of multiplying by the coordinate function x. Thus *H consists of the bounded Baire functions of a single SA operator, and by Theorem 4, CL also is such. The next result was pointed out to us by I. M. Singer. COROLLARY 5.2. An algebraic isomorphism between two commutative W---algebra3 (not necessarily on the same space ) i necessarily blcontlnuoua in the weak topology and preserves the operational calculus for bounded Baire functions. If ys is an algebraic isomorphism of the W#- algebra d onto 1 the W-x algebra Q o9 and if ty and 7% are masa algebras algebraically t~> J. 2 isomorphic to d ., and (L~ respectively, then clearly ^f-\ anc * ^o are algebraically isomorphic. This implies (loc. cit. ) that 2^ and /#C are unitarily equivalent. The corollary now follows from Theorem 4. In the case of a separable Hilbert space, the W-Ji algebra generat- ed by a SA operator consists of all bounded Baire functions of the operator. This Is no longer the case for Hilbert spaces of higher dimension. The next corollary shows however that with an appropriate generalization of the operational calculus, the result remains valid. We first make the following Definitions 6.1. A projection P on a Hilbert space is called countably-decomposable relative to a W#-algebra CL if every family of mutually orthogonal projections in CL each of which is bounded by P is at most countable. An operator TJ on a Hilbert space ~H* is called an extended Baire function of a normal operator T on "^ if for every countably-decomposable projection P in the W#-algebra generated by T 38 I* E. Segal and I, U leaves P7V- invariant, and the contraction of U to P>i^ is (in the usual sense) a bounded Baire function of the contraction of T to PW". We mention that a very explicit development of an apparently closely related notion of extended function is due to Plessner and Rokhlin CsJ , who deal with an operational calculus for functions on the reals which depend also on a variable ranging over a certain Boolean algebra. COROLLARY 5.3. The Wfr-algebra generated b a SA operator T and the identity consists f all (bounded) extended Baire functions of T. Let CL be the W#-algebra generated by the SA operator T on "#" and I, and let S be any extended Baire function of T. To show that S CL is equivalent, by a well-known theorem of von Neumann, to showing that S e o! ! , or that SU - US for all U e Q? . Now let P be a countably-decomposable projection relative to CL which is in CL Denot- ing contractions to ?H by subscribing "P", it is clear that Sp is a Baire function of T_, and that O is generated by T and I , so Sp ( Q p)". Now TU = UT and contracting to ?*& yields the equation (TU) = (UT) . As U leaves P/<^ invariant, it follows that T U ~ U p T p , or U p e (CL) 1 . It results that S p U p = U p S p , or (SU) p = (US) p , which by virtue of Lemma 2.7 shows that SU == US. Now let S be arbitrary in the algebra CL defined above . If P is a countably-decomposable projection in d , then clearly Sp 6 #p and Op is the W^-algebra generated by T p and I , so it is sufficient for the remainder of the proof to restrict attention to the case in which I is countably decomposable relative to fl. By Theorem 4, it may also be assumed that CL is mas a. Hence by Theorem 1, Remark 3.1, and 'Lemmas 1.2 and 2.5, the proof of the corollary will be concluded by establishing the following result. DECOMPOSITIONS OP OPERATOR ALGEBRAS. II 39 LEMMA 5.3.1. Let M = (R, 1%, r) be a finite perfect measure space , and let; k be_ a real-valued continuous function on M such that the Wfr-algebra generated by_ the identity and the operation on L^fM) of multiplication by_ k, is^ the multiplication algebra ojf M. Then for every real-valued continuous function h on M there is a bounded Baire function /O such that h(p) =r /o(k(p)) almost everywhere on M. j j - Let ^ be the cr-ring of all elements of ft of the form k~l(B), where B is a Borel subset of the reals. We show first that every element of 7t differs by a null set from some element of & . Let 1C be the set of all elements of 7/ ^ Lg(M) which are measurable relative to <j . By the Riesz*-Fischer theorem, 1^ is a closed linear manifold in H"> and it is easily seen to be invariant under the operation T of multi- plication by k. Hence K is invariant under the multiplication algebra of ytf . It can be shown (cf . [12] ) that any such manifold consists of all elements of ty which vanish a. e. on some set K in $ . But, de- noting the characteristic function of any set E as '), we have 'Y Si A K. then in */(? > and as the function 1 which is identically one on R is in V, 1 -X v or % v is also in "fc. This implies that % v vanishes a. iv-Jv & A e. on K, or that K is a null set. It is clear that adding a constant to k does not materially affect the situation, and hence we may assume that k(p) > 1 for p . R. Now let h be an arbitrary real-valued bounded measurable function on M, and let T and u) be the functions on the Borel subsets B of the reals defined as follows: f(B) =y E h(p)dr(p) and u) (B) = J^ k(p)dr(p), where E = k~l(B). It is easily seen that T and cJ are finite regu- lar measures on the reals. We observe now that V is absolutely continu- ous with respect to td , and in fact |T(B) | < ||h|| ^ J^ dr(p) < ||h|| ookfpjdrtp) = ||h|| ^(B). Hence there exists a bounded 40 I. E. Segal cO -measurable real -valued function y^ on the reals such that t(B) = /. V'XxJd u)(x). In a finite regular measure space, any bounded measurable function is equal a. e. to some bounded Baire function, so that "^ can be taken to be the latter type of function. We conclude the proof by showing that a. e. h(p) ir "^(kfp)). It is evidently sufficient to show that J^ h(p)dr(p) = J^ )^(k(p) )dr(p) for all E 7? but as every element of 72 differs by a null set from an element of ^ , It is sufficient to establish the last equation for E of the form k-l(B), B being Borel. For such a set E, the equation is valid by the definition of ^ . PART II. NON-COMMUTATIVE ALGEBRAS. 7. Decomposition theory. In this section we obtain a decomposi- tion of an arbitrary W*--algebra Into parts of uniform multiplicity, and in the following section we determine the structure of the general W-*-algebra of uniform finite multiplicity. Their structure for the case of infinite multiplicity remains, however, obscure, except in special cases. The present decomposition coincides in the case of a commutative algebra with that obtained in Part I, but the method used in Part I is in- appropriate in the non-commutative case, and the present method is not as well adapted to the abelian case as that of Part I. The basic difficulty in extending the method of Part I is that an algebra with a cyclic element need not be of uniform multiplicity one, In the non-commutative case, (with any reasonable definition of uniform multiplicity). The technique we em- ploy here is in part an extension to the non-commutative case of a reformu- lation of the technique employed in fsj . Definitions 71. A W*-algebra & on a Hilbert space ty is said to be of minimal multiplicity n if n is the least upper bound of the cardinal numbers m such that there exist m mutually disjoint DECOMPOSITIONS OP OPERATOR ALGEBRAS. II 41 projections P^ in the comrautor # f of Ci such that the operation of contracting CL to ? 1+ is an algebraic isomorphism. It is said to be of uniform multiplicity if it consists only of the zero operator. It is said to be of uniform multiplicity n if for every nonzero projection P in the center of Q. 1 , the contraction of Q to P^/ has minimal multiplicity n. THEOREM 6. For any Wfr-algebra # and each cardinal n > 0, there exists a projection P n in the center of ft 1 , such that the P n are mutually orthogonal and have union equal to_ I, and with the contrac- tion o d t the range o P o_f uniform multiplicity n. LEMMA 6.1. If. the minimal multiplicity oJT a Wtt-algebra CL 22. ^ i 3 . ^ n > and ; i P is. . nonzero projection in the center of Q? , then the contraction o OL to the range of, P Is likewise o minimal multi- plicity ^ n. Let (PM } 6 a family of mutually orthogonal projections in Q_ ! such that the contracting of & to P^ 1+ is an isomorphism, for each /x. . Then the PP^ are mutually orthogonal projections in ( Q ? ) ' , where Qp is the contraction of CL to P7f, and the contracting of CL to PP^u. H- is an isomorphism, for if T p e Q 9 say T p is the contrac- tion of Ted, and PP^ T = 0, then it follows that PP^ T = 0, so PT = and T. = 0. The lemma follows now from the definition of minimal multiplicity. LEMMA 6.2. Let CL be a W-algebra on /^ and A> \ be a c /^ j family o mutually orthogonal projections in Q} such that the contrac- tion of (2 to_ the range of P^ has minimal multiplicity > n, for all M Tnen the contraction of d to the range of LV P^, likewise has, minimal multiplicity > n. 42 I. E. Segal If m Is any cardinal < n, then by the definition of the minimal multiplicity there exist projections Q^v , where V ranges over a set of cardinality m, in ( Qju. ) f * where Q/ 4 - ^ s ^he contraction of (1 to P^ /^ , which are mutually orthogonal and such that the contracting of (jiu to QVv ^M ^ is an isomorphism. Now let Q^^v *>e the unique projec- tion orthogonal to I - P^u whose contraction to P^u. ^ is QVv ; it is not difficult to verify that P^ Q^uv Q/*v and that Q^v e Q} . Setting R v UM Q^cv > then R v R v f =0 if V ^ vS for Q r v Q^L. v = ( V V P " )(P v'V' v ) = " V ^ V ', and it is not difficult to verify that if for} ***& {^B^,^ are families of projec- tions on 7*^ such that A^ B-t = for any & and T , then ( U r A r )( U T B^ ) = 0. Thus {HV) is a family of m mutually orthogonal projections in (X 1 To conclude the proof of the lemma it suffices to show that the contracting of Q. to R v 1+ is an isomorphism, where Q- is the con- traction of Q. to the range of (J^ P^ . Now suppose that R V T = with T e Q . Multiplying the last equation by Q^v shows that QixvT ~ 0, which implies that Q ^ v TP^u = 0. As the contracting of d^ to QVi V P/-c W' is an isomorphism, it results that TP^ = 0, It is easy to deduce that T = 0. LEMMA 6.3. Let Q be a W -algebra on 7V and {*>u} a fam- ily of projections in the center gj* ( such that the contraction of Q^ t the range o P^ has minimal multiplicity > n f for all LL . Then the contraction of Q to_ the range of L^ut ^M likewise has minimal multi- plicity > n. There is no essential loss of generality in assuming that the in- dex set over which u varies is well-ordered with first element 1. De- fining Q^ = P^ - U v<u P v , for u > 1, and Q x = P lf it is not DECOMPOSITIONS OP OPERATOR ALGEBRAS. II 43 difficult to verify that the Q ^ are mutually orthogonal, and that Q ^ < P u. . It follows from Lemma 6.1 that the contraction of d to Q ^ ty (which is the same as the contraction to Q^ # of the contraction of Ci to P^u^) has minimal multiplicity > n, and Lemma 6.2 now implies that the contraction of & to ( {J^ Q^ )1+ has minimal multiplicity > n. !Die lemma now follows from the observation that LAiQju. ~ LJ^P^ . LEMMA 6.4. If Q, is. a W~algebra of minimal multiplicity n 2S & then there exist mutually orthogonal projections P and Q in the center of # f such that the contraction of Q to P> is. of uniform multiplicity n, the contraction of Q t Q #- i o minimal multiplicity greater than n f and with sum P + Q equal t the maximal projection in a . If d is of uniform multiplicity n, then it is obvious that tt conclusion is valid. Assuming that Q is not of uniform multiplicity n, let Q be the least upper bound of all projections R in the center of Q f such that the contraction of Q to R ff is of minimal multiplicity > n. Then* by Lemma 6.3, the contraction of d to Q ty has minimal mul- tiplicity > n, and putting P ~ E - Q, where E Is the maximal pro- jection in (3 , it easily is seen that the contraction of & to P/V' is of uniform multiplicity n. PROOF OP THEOREM. We define the P n by transfinite induction. We first put P for the orthocomplement of the maximal projection E In 0. . By Lemma 6.4, E = P -f Q where P and Q are mutually orthogonal projections in the center of d , and with the contraction of OL to Q/V* of uniform multiplicity > JL; we set P I = P. Now suppose that P m has been defined for m < n in such a way that these are mutually orthogonal projections in the center of Q} with the properties that the contraction 44 I. E. Segal of (2 t ? m^ is of uniform multiplicity m, and that the contraction of d to (I - ^m)^ Is f minimal multiplicity >> m, where 1^ ~ Ur-m V Putting N = U m<n ? n , then I - N ^ U m<n (I - Pj n U m<n (I - I^), and by Lemma 6.1 the contraction of d to (I - Nj/V' m<n has minimal multiplicity > n. If N = I we set P n , = for all n f n; otherwise we take P n = P, where P and Q, are mutually orthogonal projections in the center of Q? such that P -f- Q = N, and with the properties that the contractions of d to P7^ and to Q ^ are respec- tively of uniform multiplicity n and of minimal multiplicity > n (the existence of P and Q being assured by Lemma 6.4). It is clear that in either case the hypothesis of the induction is valid at the next stage, so that the P n are well-defined, and it is easily seen that they have the properties given in Theorem 6. (We note that the minimal multiplicity of the contraction of Q. to any invariant closed linear manifold is bounded from above by the cardinality of a set of mutually orthogonal projections in Q 1 , and hence by the dimension of 7f). We conclude this section by showing that the present notion of algebra of uniform multiplicity n agrees in the case of commutative alge- bras with the notion introduced in the first part of this paper. THEOREM 7. A commutative W^-algebra is, of uniform multiplicity n iEL the sense pJT Definition 2.1 if and only if jLt is_ o uniform multiplic- ity n according t Definition 7.1. LEMMA 7.1. If a commutative W*-algebra i 2S /^ i 2 unifoim multiplicity n in the sense of Definition 2.1 and of uniform multiplicity m iS the sense o Definition 7.1, then m = n. It is clear from Definition 2.1 that there exist n mutually orthogonal projections P^ in Q 1 such that the contracting of A to Pi 7f is an isomorphism, so m > n. Now suppose that n > *<* . Let N DECOMPOSITIONS OP OPERATOR ALGEBRAS. II 45 be a nonzero projection in Q. such that the contraction Q.*, of & to Nfa- is countably-decomposable. Then N is in the center of Q* so d^ is of uniform multiplicity n in the sense of Definition 2.1 and of uni- form multiplicity m in the sense of Definition 7.1. It is clear that there exist n separating vectors {x^} for # N such that #3*^ i orthogonal to QjjX for ^ ^ u. 1 . Now assuming m > n, let {Q v } be a family of m 1 mutually orthogonal projections in ' where m > m 1 > n, such that the contracting of Q N to Q v (N>f) is an isomorphism. The projection x,^ of x^. on QyNT'f vanishes except for countably Many V > so that the set of all indices v for which x^x v 7^ for some yu has cardinality at most ^ o n. However, for each V there is a JLA such that x^uy ^t- 0, as otherwise, taking ^ to be an index such that x^ = for all ^u , then clearly x^ is orthogonal to Q^N#- for all JJL . In particular (x^ , Ay x ) = for all yU , all A Q^ and all y x e Q X N/^, It follows as in the proof of Lemma 2.9 that y^ 0, a contradiction. Hence m < b^n = n, so m = n. To conclude the proof of the lemma it suffices to show that if n is finite, then m < n. Let N and fav} De as ^ n ^ ne preceding para- graph. Let y v be a separating vector for in Q V NW* The remain- der of the proof is essentially identical with the proof of Lemma 2.11, with y v playing the same role in both proofs. PROOF OP THEOREM. Suppose to begin with that Q. on /-/is of uniform multiplicity n in the sense of Definition 2.1, where we may take n > 1, as the case n = is trivial. Let {p } be an indexed family of projections satisfying the conclusion of Theorem 2. Then the contrac- tion of Oi to PJL ^ is of uniform multiplicity i in the sense of Defi- nition 7.1 and of uniform multiplicity n in the sense of Definition 2.1, so P^ = for i ^ n by the preceding lemma (the algebra of all 46 I. E. Segal transformations on a zero-dimensional space being taken to have uniform multiplicity n, for all n) and it follows that Q is of uniform multi- plicity in the sense of Definition 7.1. Now assume that d on 7f is of uniform multiplicity n in the sense of Definition 7.1, where again we may take n > 1. Let 0*1.1 be projections satisfying the conclusions of Theorem 2. Then the contrac- tion of Q. to P^T"/" is of uniform multiplicity i in the sense of Def- inition 7.1, so PI = for i ^ n, and it follows that d is of uniform multiplicity n in the sense of either definition. Algebras of uniform multiplicity. We show next that a W#- algebra Ct is of uniform multiplicity 1 if and only if Q} is com- mutative. The classification in Part I of commutative W#-algebras within unitary equivalence thereby induces a similar classification of algebras of uniform multiplicity 1. A W-#- algebra of uniform finite multiplicity n is shown to be an n-fold copy of an algebra of multiplicity 1. For in- finite n the corresponding conclusion is not valid (as is clear from con- sideration of the case of a factor of type II) and our results as regards this case are highly incomplete, except that a basic special case is given a full treatment in the next section. The commutative case of the next theorem is due to Nakano THEOREM 8. A W*-algebra !. of uniform multiplicity one if and only ^f its commutor is. commutative. Let d be of uniform multiplicity 1 on 7-A * so I . u. Clearly $ f is commutative if and only if Q* d GL 9 and this In turn is the case provided Q. contains all projections in Q? . Now let P be an arbitrary projection in Q} , let R be the L.U.B. of all projec- tions Q in G such that QP = 0, and let S be the L.U.B. of the projections Q in & for which QP = Q. It is easily seen that DECOMPOSITIONS OP OPERATOR ALGEBRAS. II 47 HP = and SP =s S, so RS = R(PS) = (RP)S = 0. Now If U la an arbitrary unitary operator In d 9 the equation QP = Implies that TJttQPU = = (U#QU)P. It follows that R = TI*RU, i. e v R is in the center of d . Similarly S is in the canter of CL . Now I - R > P(I - R) = P > PS = S, so I - R > P >S. By the definition of uniform multiplicity, with I-R-S =0, in which case the preceding inequality shows that P e & , or the contraction GL- of Q to the range of I-R-S is of uniform multiplicity 1. We show that the latter alternative is an impossibility. We show first that the contracting of $, to P(I - R - S)^ is an isomorphism, 1. e. f that if U e a 1 and if P(I - R - S)U = 0, then (I - R - S)U = 0. Now it is sufficient to show this for the case when U is a projection. To see this, let N = P(I-R-S), so NU = and NUU# =: 0. Clearly Nf (UU#) = for any polynomial f such that f (0) =r 0, and hence if K is any projection in the W*-algebra gen- erated by OT#, NK = 0. As this algebra is generated by the projections it contains, the equation NK = for all K implies that NUU* =r (NU)(NU)*, so NU =s 0. Now taking U to be a projection in QL , sup- pose that P(I - R - S)U =r 0. Then (I - R - S)U < R, by the definition of R, and multiplying this inequality by I-R-S (which commutes with both sides) yields the inequality (I - R - S)U 0, so (I - R - S)TJ ^ 0. Next we show that the contracting of &- to (I - F)(I - R - 3)^ is also an isomorphism. As in the preceding paragraph, it is sufficient to show that if U is a projection in Q. such that (I - P)(I - R - S)U = 0, then (I - R - S)U = 0. The equation (I - P)(I - R - S)U = can be put in the form P(I - R - S)U = (I - R - S)TJ, which by the definition of S implies that (I - R - S)U < S. Multiplying both sides of this in- equality by I-R-S shows that (I - R - S)U = 0. Now the projections P(I - R - S) and (I - P) (I - R - S) are obviously orthogonal, so what 48 I. B. Segal has just been proved shows that Q ^ Is of minimal multiplicity at least 2, a contradiction. Hence the second alternative above was impossible* Now suppose conversely that Q f is commutative, so Q f C d If Q is not of uniform multiplicity 1, then for some nonzero projection P in the center Q 1 of # f the contraction Q^ of & to P 1+ is of minimal multiplicity at least 2. Suppose then that Q and Q are mu- tually orthogonal projections in ( ^L) 1 such that the contracting of d . to Q'P?"/' is an isomorphism (i = 1,2 ). Let Q and Q be the unique projections on #- which coincide on PH- with Q* and Q* and which annihilate the orthogonal complement of ?>/* Then Q-^ and Qg are in # f , for if T is arbitrary in Q., then taking the case of Q^ we have (noting that Q^p := PC^ = Q^ C^T = (C^PJT = (^(PT) = (^(PTP) (for P is in the center of a ) = (PTP)^ (as Q ( O^ 1 ) = (TP)Q 1 = TtPQ-j^) ~ TC^. Therefore (^ and Q^ are in GL 9 and as Q and Q are mutually orthogonal, (Q^P)(QgP) =: 0, which shows that the contracting of Q.-^ to Q-jP^ annihilates Qo^* *&& so is not an isomor- phism. Algebras of uniform multiplicity 1 play a conspicuous role in the following, and so it is convenient to make the following definition, which is justified by the fact that a W#-algebra containing I is of uni- form multiplicity 1 if and only if the lattice of all closed linear sub- spaces invariant under the algebra is a Boolean algebra* Definition 8.1. An algebra of operators is called hyper-re due ibla if it is a W^-algebra of uniform multiplicity 1. COROLLARY 8.1. If two hyper- reducible algebras are algebraically isomorphic, then they are unitarily equivalent. Let the hyper-reducible algebras 0-^ and # 2 on Hilbert spaces H and A/- 2 b algebraically isomorphic via the map 9 P on Q.\ DECOMPOSITIONS OP OPERATOR ALGEBRAS. II 49 to $ 2 . Then <p maps the center C. of Q ^ isomorphloally onto the center ^7 2 of # 2 . Let /((-^ be any masa algebra on ~H\ which con- tains C^. As Q^ is hyper-reduoible, C -= (#.), so ()' =: (<?!)" - # lf and hence 7f^ d Q^. Putting *f g = <?( ^J, then 2f 2 is masa on 7^, for as ^ is an algebraic isomorphism, tyL is max- imal abelian relative to # 2 , i. e., (^fg) 1 " # 2 = ^ 2 - Taking commut- ors, it follows that ~>t ^ u Q^ = (7^ 2 )t f and as (# 2 ) f = Cg d 7^ 2 , it results that ^ 2 = (tygJS i. e v V is masa. We can now apply the result in fioj that if two masa algebras on Hilbert spaces are algebrai- cally isomorphic, then there is a unitary transformation U between the spaces which implements the isomorphism. As ", is carried onto C by the opera tor- isomorphism induced by U, ( C. ) t is carried onto ( O ) by ^ 2 that isomorphism, i. e.^ ^ is mapped onto ? 2 . COROLLARY 8.2. A W*-algebra finite \iniform multiplicity n>0 i& ^mitarily equivalent to. an n-fold copy o a hyper-reducible algebra, the latter algebra being unique within unitary equivalence. Let CL be a W*-algebra on & of finite uniform multiplicity n, and let P^,.tP n be an indexed set of n mutually orthogonal projec- tions in d 1 such that for each i, the contracting of Q. to P. 14^ is an isomorphism. We show first that the contraction Q^ of Ci to P /"f is hyper-reducible. Let Q f be a nonzero projection in the center of 2.^, and let Q be the projection in Ci of which Q f is the contraction. As the basis of an indirect proof suppose that the contraction ~& of & to Q'P^M has minimal multiplicity > 2 (for some fixed 1), so that there exist projections R f and S 1 on Q f P^^ In ^3 f , mutually orthog- onal, and such that the contractings of t3 to R'Q'P^^" and to S'Q'P^^ are isomorphisms. Putting R and S for the projections on QM which agree on Q'Pj^ with R 1 and S 1 respectively, and which annihilate 50 I. E. Segal i)^, then It is not difficult to verify that R and S are in <3, where C? is the contraction of Ci to QM, that R and S are mutual- ly orthogonal, and that the contractings of C to RQ7V- and to SQ7^ are both isomorphisms. Now the contractings of O to P-QM (J 96 i) are likewise isomorphisms and the contractions of P* to Q TV- (j ^ i), R and S, together constitute asetof n + 1 mutually orthogonal projec- tions in C . It follows that C is of minimal multiplicity at least n 4- 1, but Q is in the center of d 9 so this is in contradiction with the definition of uniform multiplicity, which requires that C be of uni- form multiplicity n. By Corollary 8.1, there is a hyper-reducible algebra ~fS such that each Q. ^ is unitarily equivalent to ^3. It is straightforward to show that d is unitarily equivalent to an n-fold copy of 13 The final theorem in this section concerns the uniqueness of multiplicities in the non- commutative case. THEOREM 9. An m-fold copy of a hyper-reducible algebra f i unitarily equivalent to_ an n-fold copy ojT a hyper-reducible algebra C i and only 1 m = n and E and C are unitarily equivalent* As the "if" part is clear, we assume that the W*- algebra CL on ty is unitarily equivalent to an m-fold copy of B and also to an n-fold copy of <7 Then Q, is algebraically Isomorphlc to both & and C , so that & and C are algebraically isomorphic, and hence (by Corollary 8.1) unitarily equivalent. We can now assume that j& = C ; for each T e C , let f (T) and y(T) be the unitary transforms in d of the m- and n-fold copies of T, respectively. Now let ?0f be a masa subalge- bra of jy (the existence of which is shown in the proof of Corollary 8.1), let <p (W fc C , and let )^-l( C } = % . Then 7? is maximal abelian and SA in J& 9 as <p and yr are algebraic isomorphisms, and hence masa DECOMPOSITIONS OP OPERATOR ALGEBRAS. II 51 (of. the proof of Corollary 8.1). It follows that ^ and >/ are uni- tarily equivalent jjLoJ .As C is unitarily equivalent both to an in- fold copy of y*( and to an n-fold copy of >f , it results from Lemmas 2.9 and 2.11 that m = n. COROLLARY 9.1. Let [> n ] be a family of projections in the center of the W -algebra Q on /e indexed b cardinal numbers n, such that for each n the contraction of Q to the range ojF P n has uniform multiplicity n, and (J n P n ~ I * Then 11 { p n } i * family with the property given in Theorem , for all finite n P = P f (so that if //- is separable, this equation is valid for all n). LEMMA 9.1.1. If d is. a W*-algebra on #- , and i R and S are projections in the center of A sucn that the contractions of Q, to R#- and S^ have uniform multiplicities m and n respectively, where m ^ n and a_t least one of m and n is_ finite, then RS ~ 0. Clearly the contraction of (2 to RS>^ has uniform multiplici- ties both m and n, and if these are both finite, it follows directly from Theorem 9 and Corollary 8.2 that RS = 0. Now let n be finite and m infinite, and suppose that RS ^ 0. Then the contraction $ o of t to RSfa^ is unitarily equivalent to an n-fold copy of a hyper-reducible algebra ^ . Let ^ be a masa subalgebra of J& , let <p be the alge- braic isomorphism of & onto Q. o induced by the map of f onto its n-fold copy and by the unitary equivalence of this copy with $ o , and set <jp(^) =; 7( , so that ^ is unitarily equivalent to an n-fold copy of 7^. Now as d is of infinite uniform multiplicity, there exist mutually orthogonal projections P^, Pg, ... in & f such that for each i the map T ^P^T is an isomorphism on & . As d => % , <2 f <=. >(' so that the P 1 are in 7 f , and clearly the map T ^ P^T is an isomorphism on 7( . Thus 7f is of minimal multiplicity N^. On the other hand, it 52 I. E. Segal is of uniform multiplicity n according to Definition 2.1, hence of uni- form multiplicity n according to Definition 7.1, so that it cannot be of minimal multiplicity ^^ . PROOF OF COROLLARY. Let P n be the L.U.B. of all projections Q In the center of Q. such that d has uniform multiplicity n on by Lemma 6.3, d has uniform multiplicity n on P n 7^ Let P^ = U P and PJ = (J P'. Evidently P' < P for all n and n ^ v o n n ^ >< n n - n P^ < PQQ . By the preceding lemma, the P n with finite n are mutually orthogonal, and are also orthogonal to the P m with Infinite m, and hence orthogonal to POO . As ( U n< ^ ?;) - P^ - I - ( U n< ^ P ft ) " ^ > we have P^ = ? n for n < K and P^ - P^ . Finally, if 7^ Is separable, it is plain that necessarily P n zr for n > >f . 9 * Algebras of type 1^* We give a structure theorem for W#-alge- bras of "type I", where these are, roughly speaking, algebras which are direct integrals of factors of type I. At the same time we obtain a com- plete set of unitary invariants for such algebras, which can be regarded as fully known by virtue of this classification. The following definition of algebra of type I is equivalent to a definition in l] for certain ab- stract algebras. Definition 9.1. A W#-algebra is said to be of type !T If every projection in &' is the least upper bound of projections of type II in ft 1 , where a projection P In # f is of type M If whenever Q is a pro- jection in Q. 1 such that Q < P, then Q = RP where R Is a projection in a a*. Roughly speaking, a projection P is of type M if when the alge- bra is decomposed as a direct integral of factors, P decomposes into an integral of projections each of which has a range which is of (ordinary linear) dimension at most one, so that it is akin to a minimal projection, DECOMPOSITIONS OP OPERATOR ALGEBRAS. II 53 as our terminology is intended to suggest. The following theorem asserts essentially that a W#-algebra is of type I if and only if it is a direct sum of n-fold copies of hyper- reducible algebras. THEOREM 10. A W*-algebra Q on ^ is_ of type I. if and only If there exists a family ^P n ~} L projections in the center of Q* , indexed by cardinal numbers n, which are mutually orthogonal and have union I 9 and are such that the contraction of Q tio P 7f (n :> 0) is^ unitarily equivalent to an n-fold copy of a hyper-reducible algebra JCT^ of uniform multiplicity n, while I-P i the maximal projection in CL The P n are unique as. are , within unitary equivalence , the ^ n Conversely Q Is^ determined within unitary equivalence by the knowledge 2 the jy n as abstract algebras (which knowledge determines the ^ n within unitary equivalence) , or alternatively by the unitary invariants o the commutative algebras jj ^, together with the dimension o the range o V Remark 91. Naturally a second alternative to the S" n as uni- tary invariants are the invariants of the & ^ as given in Part I. This implies that a cardinal-valued function P(n,m,p) of triples of cardinal numbers could also be used in place of the unitary-equivalence classes of the f R . Here P(n,m,p) is the cardinality of the number of summands isomorphic to the measure ring of the product measure space 1*^ (where I is the unit interval under Lebesgue measure) which occur in a decomposition into finite homogeneous parts of the measure ring for multiplicity m of {^9 and it is necessary as earlier to normalize P(n,m,p) f say by setting it equal to 1 for the case when p ^ and the number P of summands Is determined only within the Interval 1 ^ P < <% and I must be suit- ably interpreted. Clearly any cardinal-valued function P(n,m,p) which vanishes for sufficiently large n, m, and p, then corresponds to a unique 54 I. E. Segal unitary equlvalenoe class of W* algebras of type I containing the identity operator. The first six of the following lemmas are needed in the proof of the "only if" part of the theorem, while the remaining lemmas are for the "if" part. LEMMA 10.1. Let T be a non-negative SA operator in the W-a- alge- bra CL 9 and suppose that T < P and FT = TP ~ T, where P is_ a pro- jection of type M in CL . Then there exists an operator S in the center 2 0- auch that T = SP and < S < I. The W#-algebra C generated by P and T is obviously commu- tative and has a unit, and so is Isomorphlc to the algebra C^(P) of all continuous functions on a compact Hausdorff space \ . It follows without difficulty that there exist projections Q in in C and real numbers <*.. (i,n = 1,2,...) such that for each fixed n, the Q^ n are mutually orthogonal, the <* satisfy the inequalities 0< &-^ n <: 1, and with T n - >T uniformly where T n = ^~*iSn^in and T n ~ T +1* Evidently Q^ < P so that Q^ n = R in p vltti T n = ( 2.*^ ^in^in^ or ^n == ^rf with R & n &* * Now let S n be the G. L. B. of all such non-negative operators R^; then clearly T n ~ S n P* Now S n < S n ^ for denoting lattice intersection in the sat of SA elements in C by "A", we have T n ~ T n A T nfl S n P x\ S nf ^ ~~ (S n A S n ^ 1 )P (noting that S n P =: S n A p) which shows that S n A s ntl Sjj, by the minimality of S n , or S n < S ntl . It follows that {s n } con- verges strongly to an operator S in A n A 1 and T SP; as it is eas- ily seen that < S n < I, we have < S < I. LEMMA 10.2. Anj projection of^ type M in a W-algebra is contained in a maximal projection of type M. DECOMPOSITIONS OP OPERATOR ALGEBRAS. II 55 This follows at once from 2orn f s principle once it is shown that the L. TJ. B. of any chain of projections of type M is again of type M. Now if *y is such a chain in the W*-algebra 9 and Q is the L. U. B. of the elements of 3^ suppose that A is a projection in & such that A < Q. To conclude the proof of the lemma it is only necessary to show that A = RQ for some R in C ^ &. Now for any P *f we have PAP < PQP = P. It follows from the preceding lemma that PAP :=. RpP for some operator R in , n C 1 such that < Rp < I. Putting Sp for the G. L. B. of all such Rp, then clearly PAP = S p P f and S p is a monotone Increasing function of P, for if P^ e *3* (i = i,2) and p l i P 2* then ^l**! 3 ^ ~ P 2 S P AP 2 or *2**2 S P P 2 whlon implies S p < S p . Now let S be the L. U. B. of the S - P e T. Then S is 2 "" r l p the strong limit of the (Sp} where this indexed set is directed by the usual ordering on the indices, and passing to the limit in the equation PA? = S p P yields the equation QAQ = SQ, or A = SQ. LEMMA 10.3. If P and Q are projections of type M in the W- algebra C 9 and if R and S are mutually orthogonal projections in C * T f , then RP SQ is. also of type M in C Suppose that E is a projection in <C such that E < RP - SQ. Then E<R + S so E = ERi-ES, and ER < RP so ER = R'RP with R f e C * C f . Similarly ES n SSQ with S' e C C f . Thus E = R'RP * S'SQ =r (RR t SS')(RP f SQ). LEMMA 10.4. If d i a W^-algebra of type I on /f and if JL, is any closed linear sub space invariant under , then the contraction of I. Let Q-^ be the contraction of d to and let P-^ be a nonzero projection in (<2i) f . Let P be the projection on /^ which 56 I. E. Segal extends P and annihilates ft- Q . Then P e fit for If U ^ <2 , we have for x f UPx = Ux = PUx, and for x e # ,, UPx = = FUx, so UP ~ PU. Let Q be any nonzero projection of type M in d f such that Q P. Then Q annihilates ^ . , so it leaves o& invar- iant; let Q. be its contraction to o . Then Q^ . (d3^)' for if U^ 2^, say UL is the contraction of U d , then we have for x c , QiU-jX = QUx = TJQx = U-^x. Also it is clear that Q^ < P^. Fi- nally Q x is of type M for if S^ is a projection in ( &^)' , then S^ is the contraction to d of the projection S in d f where S(^ G * ) r: 0, and the equation S^ < Q 1 implies that S < Q; and as Q is of type M, it results that S = RQ with Re Q r\ QI 9 from which it re- sults that S-j^ zr f^Q-j^ with E^ e QT (Q..) 1 . LEMMA 10.5. Let # ^2. , W^-algebra on /V' of type I, and let P be a maximal projection of type M in (2} . Then the contracting of Q to_ P/V" i an isomorphism. It is sufficient, by a remark in Part I, to show that if S is a projection in d such that PS = 0, then S = 0. Now let T be the L. U. B. of all the projections S in & such that SP = 0. If SP = and U is unitary in &, then U*-(SP)U = so that (U#SU)P = 0. Thus T is the L, U. B. of a set of projections which is invariant under the inner automorphisms induced by the unitary operators in Q- , and hence is in the center of d . If Q is a projection in # of type M and such that T > Q, then by Lemma 10.3, P(I-T) + QT = P * Q is again of type M. By the maxlmality of P, Q = 0, and since T is the L. U. B. of such Q, it follows that T = 0. LEMMA 10.6. If d ! a nonzero W*- algebra on 1+ of type I, then there exists a nonzero projection P In Ci Q. 1 such that the contraction of fl to P H- l unitarlly equivalent to an n-fold copy of a DECOMPOSITIONS OP OPERATOR ALGEBRAS. II 57 hype r-re due ible algebra, for some cardinal n > 0. Let ^ be an Indexed family of projections P^ In fl 1 which is maximal with respect to the properties: 1) for each JUL the contracting of d to ?ju W is an isomorphism; 2) the P^ are mutually orthogonal; 3) each P.^ is of type M (the existence of *y is clear from Lemmas 10.2 and 10.5). Let Q^ be the L. U. B. of the projections Q in CL for which Q(I - U^ Pyu ) = 0. Then Q^ ; 0, as otherwise I - LL ^ 7^ 0, and the contracting of d to (I - U^P^ )H is an isomorphism. In continuation of this argument, let d^ on M'^ be this contraction; by Lemma 10.4, Q is of type I. Let Q^ be a maximal projection in (^2- ) f of type M, and let Q be the projection on ^ (necessarily in & f ) which extends Q^ and annihilates M Q W^. Then the contracting of d to Q W ( - Q^ "H\ ) is an isomorphism, Q is orthogonal to all the FU_ , and it follows as in the proof of Lemma 10.4 that Q is of type M. This contradicts the maximality of *3< and so shows that QQ ^ 0. By an argument used in the proof of Lemma 10.5, Qo is in the center of # . Putting P^ =. Q^?^ and K^ = P^ 1+ , then: (a) (L leaves 7t^t invariant and its contraction d^. to 7-^ is of type I; (b) the identity in ( &^/) f (i.e. the contraction of P^ to /t^u ) is of type M in (Ct^.) 1 (by an argument used in the proof of Lemma 10.4); (c) the contracting to ^ of the contraction of d to QQ/^ is an isomorphism. It is clear that the identity operator is of type M in a W#- algebra only if the algebra is commutative, and hence the d^. are hyper- reducible. If n is the cardinality of the index set, it follows from Corollary 8.1 that the contraction of d to QQ^ (= U/* ^^ ) is uni- tarily equivalent to an n-f old copy of any one of the &/*- . LEMMA 10.7. A commutative W^-algebra is. of type I. Let d be a commutative nonzero W#-algebra on 1+ . To show 58 I. E. Segal that Ci is of type I it suffices to show that for any projection P in #', there is a projection of type M in Q* which is contained in P. Let W be a countably-decomposable projection in d such that WP ^ (cf. Lemma 2.7), let V be the maximal projection in Oi such that VWP 0, and put Q.^ for the contraction of & to 7=^ = (I-V)W7A As & is countably-decomposable and as the contracting of Q.. to f^t i &* 1 isomorphism, there exists a separating vector x^ for A^ in pM. Put- ting X M} ** or aji indexed family of separating vectors for &~ which is maximal with respect to the properties that it contains x^, and that Qjip, is orthogonal to Ax v for yU ^ V , a repetition of the con- struction at the beginning of the proof of Lemma 2.6 shows that there exists a nonzero projection Q^ in Q.-^ such that if $ 2 is the contraction of Q_ ^ to Q #1 , and if R,^ is the projection of Q. M. onto the closure of QgXyu. i then Q. = L/M- R^ and the contraction of d to R^ /A is masa, flg being unitarily equivalent to an n-fold copy of this mas a algebra, where n is the cardinality of the index set. Putting S for the projection of 'H' onto %Qi^i then it is easily seen that S < P, and we conclude the proof of the lemma by showing that S is of type M in fl 1 . Let QQ be a projection in Q whose contraction to /V- is Q., and set Q ~ (I-VjWQQ, so Q e d and Q7V- = *i"^i Now ^ leaves Rl1+ invariant, for if U is arbitrary in Q. , U = UQ + U(I-Q), and UQ leaves R^^ invariant because its contraction to Q/V' is in dg while TJ(I-Q) annihilates K^1^; therefore S e GL\ 9 Finally if T is any pro- jection in Q. 1 such that T S, then the contraction T-^ of T to QtV' is in Q and T < R . As Q is masa on R H and leaves T Q //- invariant, T X = R^^ with ^ e (3 2 n (Q ) f , and putting U Q for a projection in Q whose contraction to Q// is U- and TJ = QU- , it is not difficult to verify that T = UR, and U e Q n Q. 1 for U ^ and Q, is commutative. DECOMPOSITIONS OP OPERATOR ALGEBRAS. II 59 LEMMA 10*8. Let the W-algebra 0. be_ an n-fold copy of a hyper- reducible algebra. Then Q_ Is, of type I. It is easily seen that if {P n } Is an Indexed family of rnitual- ly orthogonal projections in the center of r W->-alebra d such that the contraction of Q. to the range of P n is of type I and with (J P - I, then d is itself of type I. It follows without difficulty from Theo- rem 2 that it suffices to oonsJder the case in which the hype r-ro duo ible algebra J^ of which d is an n-fold copy has its commutor equal to an m-fold copy of a raasa algebra ^ on Y- . Let d act on 1^ 9 J& act on d. , and let {^u } bo the copies of * which are (mutually orthogonal) subspaces of //'. If T io any operator on # and P^u. is the projection on J!^ , then P v TP^u maps o6^c into <^ v and so induces in a natural fashion an operator Tu-v on L ; by the matrix of T we mean thia matrix (CS^* )), which is a function on the direct product of the index set with itself to the operators on < . It is not difficult to verify that a matrix ((S^^, )) is the matrix of an operator in d f if and only If: (1) it is the matrix of a bounded operator, i. e v if x is arbitrary in ^ * if X M is the corresponding element of o^. and x-u the element of corresponding to x^ t then the sum -*L*. S^ v x/x is convergent (thia is the case if and only if |jSa,v/l is bounded for each fixed v as a function of j* ) and 2I V U <"yu S^v x^ II 2 is a convergent sum; (2) S^v e K for each JU , V . We omit the details of the verification of this, as they in- volve standard methods. Now let <p be the algebraic isomorphism of J3* onto ^f whldh takes each operator into the operator of which is the m-fold copy. Let /f on Jh be the n-fold copy of >f on k' . If S e &' , and ((S^v )) is the matrix of S, then Sv e &* and ((f (Syuv ))) is a matrix 60 I. E. Segal of operators in ty ; we shall show that is the matrix of an operator <|(S) In 7^ T (relative to the decomposition of the space f on which 7L acts as an n-fold direct sum of copies of ft ), and that <> Is an algebraic isomorphism onto. To show that (( jz*(S^iv ))) is the matrix of an opera- tor In ^f , It is only necessary to check condition (1) above. Now each dL^jL is in a natural fashion an m-fold direct sum of copies ou<f ^ a. For each f*. 9 let & ^ be any index in the range of values of cr , and for each y in ft let y be the element of 1-C corresponding to the projection of y on the yuth copy of 1^ in Af , and further let ?7 (y) be that element x in /^ such that x . a = y^ and x.,.^ = if ^7^ G"-, where x^o- is the element in ft, corresponding to the projection of x on the <^th copy of "ft in the yuth copy of <& . Then as (1) holds and as // 52* (S v ) )| = // s /^v |) because ^ is an algebraic Isomorphism, and ^^S^^ ( ~ ->u(S uv )yv II 9 it follows that the condition corresponding to (1) holds in the case of the matrix (( ^(S^uy ))), so that (( ^(Syw.v ))) is the matrix of an operator in ty ' Now if ((SV^v )) Is the matrix of an operator S 1 in 7f f , to T V 7 II 5^ y-1/ II 2 show that (> is onto we need only show that Z.\ v II 4L*up> (S^y* Jx^ // is a convergent sum. If Xu* is as In the preceding paragraph, we ob- i II V 7 ^-1 x /J 2 serve that (^-^(S^v, )x ^ ^ = S^ v x^c- , so I/ ->>ju<p (SJ^v )x^ If ^T 7 II V 7 ^ i IIP , It follows that Z_i v (l Z-^., P" (S' v Jx/x || ZJIZ^s^x^ II 2 = S^Z, //Z^s^v x>Mr I/ 2 < ^\\^ <r II 2 = IIS/I 2 H o.^ llx^o- II * = HSU 2 Hxll 2 . To see that (J> Is a homomorphism, observe that It is a homoinorphism on the subalgebra of operators whose matrices have only a finite number of nonzero coordinates, because & is a homomorphism, so that by the valid- ity of the standard rules for matrix operations and the strong density of DECOMPOSITIONS OP OPERATOR ALGhBRAS. II 61 that subalgebra (cf . [s] > P 137) it Is a homomorphism on the entire algebra. Thus >f and #' are algebraically isomorphic. Now > is of type I, as it is commutative, so every projection in 7( is the L. U. B. of projections of type M. As L. U. B.s of projections and the concept of projection of type M are preserved under algebraic isomorphisms, it follows that every projection in Q. 1 is the L. U. B. of projections of type M, - i. e.> d is of type I. PROOF OF THEOREM. Let & be a W*-algebra on 1+ of type I. From Lemma 10.6 it follows readily that there exists a family {p^ j- of mutually orthogonal projections in the center of Ct such that the contrac- tion of d to P^ H is unitarily equivalent to an n(/x)-fold copy of a hyper-reducible algebra, and Uyu P^ is the maximal projection in ^ . The proof of Lemma 2.2 shows that if iQv} are mutually orthogonal pro- jections in the center of a W-*-algebra C such that the contraction of to Q v V4- is hyper-reducible, then the contraction of to ( Uv Qv )*+ is likewise hyper-reducible. Putting Q^ = U n (u)=m p / x it follows that the contraction of Q to (^ 14- is unitarily equivalent to an m-fold copy of a hyper-reducible algebra, and clearly the (^ are mutual- ly orthogonal and \J m Qjn is the maximal projection in $ . Conversely, let the Wtf-algebra d on /^ be such that there exists a family {P n } of projections in the center of Q* and such that I-P is the maximal projection in d 9 the contraction of d to P n # is unitarily equivalent to an n-fold copy of a hyper-reducible algebra. To show that d is of type I it suffices to show that any such n-fold copy is of type I, and this is the statement of Lemma 10.8. Finally, the unique- ness part of the theorem follows without difficulty from Theorem 9. COROLLARY 10.1. II* a W -algebra is. of type I, then sjo is. its commuter* 62 I. E. Segal It Is easily seen that it suffices to prove the corollary for the case in which the algebra Q_ in question is an n-fold copy of a hyper-re- ducible algebra f . To show that $ ! is of type I is to show that every nonzero projection in Q contains a nonzero projection of type M. Now j? 1 is commutative so, by Lemma 10.7, every nonzero projection in J^ contains a nonzero projection of type M, and as f and d are algebrai- cally isomorphic, it follows that the same is true of OL . COROLLARY 10.2. A W- algebra type IE is^ algebraically isomor- phic Jbo a hyp e r - re due ib 1 e algebra via a mapping onto the hyp e r - re due ible algebra that is, weakly continuous and preserves the operational calculus for normal operators. The proof of this is a slight modification of part of the proof of Theorem 5. The next result is essentially equivalent to the non-trivial part of Theorem IV of CsJ . COROLLARY 10.3. (Murray and von Neumann). A factor whose com- muter contains a minimal projection *is_ unitarily equivalent t an m-fold copy o.f the algebra of all operators on an n -dimensional Eilbert apace., for unique cardinals m and n. Let 3* be a factor on 14- whose commutor contains a minimal projection E, and let P be an arbitrary nonzero projection in *y. Then the L, U. B. of TJ-^EU as U ranges over the unitary operators in *y f is a projection in the center of * f , and so equals I. It follows that the operator T = PU-frETJP is nonzero for some U, as otherwise it is easily deduced that P =. 0, but T is SA and non-negative so that^it can be uniformly approximated by linear combinations of projections Q such that odQ < T for some <x > o. Now T = U#EU, from which it follows DECOMPOSITIONS OP OPERATOR ALGEBRAS. II 63 that Q vanishes except on the range of U#EU, ao Q <- UtfEU; and as E is minimal, so also is U#ETJ, and it follows that Q = U#EU. It results that T is a scalar multiple of U**EU and it is easy to see that T =* U*EU. Hence is of type I in the sense of Definition 9.1. As the center of ~$ is trivial, it follows from the theorem that it is unitarily equivalent to an m-fold copy of a hyper-reducible alge- bra Jff on ft; . Now f* contains the corresponding copy of iJ ' on f , and Jff* c: f , so J 1 y f contains this latter m-fold copy. It follows that f* consists only of scalars, so Jfr is the algebra of all operators on % . That m and n are unique is clear from the uniqueness part of the theorem. Remark 9.2. In view of the special r&le in the foregoing of com- mutative \Y#- algebras of uniform multiplicity we should mention a relatively concrete form for such. If d on /^ is the n-fold copy of a mas a alge- bra )/( on 1"C 9 then >^ can be taken as the multiplication algebra of a measure space M = (R, 7?, r) which is the direct sum of the finite per- fect measure spaces M x =. (R^ , /? x , r x ); //* can plainly be taken as the collection of all indexed families f (p) of functions on R with JJ(. ranging over an index set A of cardinality n, and such that f^ ( . ) 6 Lg(M) for all JU and with the sum "/- 1 " f/u n ^ convergent; and Q. is then the algebra of all operators A on 7^ of the form (Af)^ (p) ~ k(p)fyu (p)> where k is a bounded measurable function on M. Now let o be a Hilbert space of dimension n. A function T(.) on M to the (bounded) operators on <, is called strongly measurable, if, whenever m{.) is a strongly measurable square -integrable function on M to > (1. e. m is n. e. the limit of a sequence of simple functions, and l(m(p)ll 2 is integrable on M), then the function n defined by the equation n(p) = T(p)m(p) is again a measurable function on M to -; and T(.) is called bounded in case //T(p)|| is bounded as p ranges over R. Now regarding 64 I. E. Segal cO as the space of all complex-valued functions f ^ on -Q. such that the sum S \JJL I f ^ I 2 ie convergent and with (f , g) = 2-* ^ f g.^ for any two such functions, there corresponds to any bounded strongly measurable function T(.) on M a (unique) operator T on 1^ such that for any f and g in 1+ , (Tf, g) = f (T(p)f (p), g (p)) dr(p), where n Lt (.., JL indicates an inner product in o ; this follows readily from the observation that the integral exists and defines a continuous function of f and g which is linear in f and conjugate linear in g. We can now state: The operator T Is. in Q f , and every element of Q ! has thi.s form, .1 . . -, for every T in Q* there ig a bounded measurable function T(. ) on M t such that for any f in -#- , (Tf)^(p) = T(p)(f (p)). If S and T are elements of Q f to which correspond in this fashion the functions S ( . ) and T ( . ) , and if <x i a complex number, then the following equa- tions are valid n. e. on M: (S+TMp) = S(p) * T(p) f (ST)(p) = S(p)T(p), (<*S)(p) = MSCp)), and S#(p) = (S(p))^-. A similar result is stated for the case when and ft are separable in fej^q* v. The only point which offers any difficulty is the fact that every element of Q f has the stated form. Now as shown above there is a natural correspondence between Q 1 and appropriate n x n matrices over ty Let the element T of Q* correspond to the matrix ((T^^ )), where /X, V -O. and T^i v is the operation of multiplication by k^ v on Lg(M), where k v can be taken to be bounded and continuous on each 7?f . The boundedness of T means that H v II 5I r T^ v f^, II 2 < II T II 2 21^ llf^ II for any f e W" , or J( H v l^k^ (p)*> (p) I 2 ) llTll 2 J SulfiA (p) I 2 dr(p). As this equation remains valid when each fyx is multiplied by the characteristic function of a measurable set, it results that n. e. on M, x^ v I Z.1^ k^ v (P)^ (p) I 2 ^ (p) | 2 . In particular, if f u, vanishes except for a DECOMPOSITIONS OF OPERATOR ALGEBRAS. II 65 finite set F of ^U f s, and if for each yu , f ^ is continuous on each R x , and if G is any finite set of indices we have HI ve G. I ^ ucF k /v (p)*i (P) I 2 1 II Til 2 21 I ** yu (?) I , n. e As both sides of this inequality are continuous on each R x , and as the complement of a null set in a finite perfect measure is dense in the space, the inequality is valid for every p e R, and since G is an arbitrary finite set of indices we can conclude that (P ) I 2 II T II 2 21 I f u (P ) I 2 f or a11 p e. R. Hence there exists a (bounded) operator T(p) on ^L with matrix ((k^uv (p))) (relative to the obvious basis), with llT(p)|| < ||T|| , and such that T(p)f^(p) = (Tf)^(p), for f as above. It is readily seen that for any f e /-f there is a sequence (f j in /4- , each f* being as above, - so that T(p)f i (p) = (Tf^) (p), - and such that f 1 converges to f in 14 ; for nearly all p, f^(p) >f^(p) in dC , and with (Tf 1 )^?) MTfMp) in ^ for nearly all p. It follows that fo] all f in "?-/-, (Tf)Jp) = T(p)f^(p) n. e. on M, and T(.) is a bound- ed strongly measurable function on 7^ to the operators on <*o . 66 I. E. Segal REFERENCES 1. I. Kaplansky, Projections in Banach algebras. Ann. Math. 53(1951) 255-249. 2. G. W. Mackey, A theorem of Stone and von Neumann. Duke Math. Jour. 16(1949) 313-326. 3. D. Maharam, On homogeneous measure algebras. Proc. Nat. Acad. Sci. 28(1942) 108-111. 4. N. H. McCoy, Rings and ideals. Baltimore, 1948. 5. P. J. Murray and J. von Neumann, On rings of operators. Ann. Math. 37(1936) 116-229. 6. H. Nakano, Uni tar invar ianten hypermaximale no male Operatoren. Ann. Math. 42(1941) 657-664. 7. , Unitarinvarianten im allgemeinen euklidischer Raum. Math. Ann. 118(1941/43) 112-133. 8. A. I. Plessner and V. A. Rokhlin, Spectral theory of linear operators. II (in Russian). Uspekhi. Mat. Nauk (N.S. ) 1(1946) 71-191. Cf. Math. Rev. 9(1948), p. 43. 9. I. E. Segal, Decompositions of operator algebras. I. Submitted to Memoirs Amer. Math. Soc. 10. , Equivalences of measure spaces. Amer. Jour. Math. 73(1951) 275-313. 11. , Postulates for general quantum mechanics. Ann. Math. 48(1947) 930-948. 12. , A class of operator algebras which are determined by groups. Duke Math. Jour. 18(1951) 221-265. 9 13. J. von Neumann, Uber Punktionen von Punktionaloperatoren. Ann. Math. 32(1931) 191-226. 14. P. Wecken, Unitarinvarianten selbstadjunjierter Operatoren. Math. Ann. 116(1939) 422-455. M E M O I R S OF THE AMERICAN MATHEMATICAL SOCIETY 1. G. T. Whyburn, Open mappings on locally compact spaces, ii, 25 pp. 1950. $0.75 2. J. Dieudonnc a On the automorphisms of the classical groups^ with a supplement by L. K. Hua. viii, 122 pp. 1951. I 90 3. H. D. Ursell and L. C. Young, Remarks on the theory of ptime ends. 29 pp. 1951. J.OO 4. K. Ito, On stochastic differential equations. 51 pp. 1951. 1.00 5. O. Zariski, Theory and applications of holornorphic junctions on algebraic varieties over arbitrary ground fields. (?0 pp. 1951. 1.40 6. K. L. Chung, M. D. Donskcr, P. Erdos, VV. H. J. Fuchs, and M. Kac, Four paper* on probability, ii, 12 + 19 -f 11 + 12 pp. 1951. 1.20 7. R. V. Kadison, A representation theory for commutative topologictil algebra. 39 pp. 1951. 1.00 8. J. W. T. Youngs, The representation problem for Frechet surfaces. 143 pp. 1951. 1.80 9. I. E. Segal, Decompositions of operator algebras. I and II. 67 + 66 pp, 1951. ^ 00 The price to members of the American Mathematical Society is 25% less than list.